Scientific Research 2010



Transfer Kernel Types of Metabelian 3-Groups

over Quadratic Fields


Statistics of second 3-class groups G = Gal(K2|K)
over complex quadratic fields K with 3-class group of type (3,3)
and discriminant -106 < d < 0.



K2 denotes the second Hilbert 3-class field of K.
The number of these fields is 2020.

The table gives the number of occurrences of each transfer kernel type (TKT) of G
for relative intervals -N*104 < d < -(N-1)*104.
"Two" denotes the number of two-stage towers of 3-class fields.
Boldface rows contain data for absolute intervals -N*104 < d < 0.

N Second Maximal Class Lower Class Tot
Two D10 D5 H4 G19 E6/14 H4↑ E8/9 G16 E6/14↑ H4↑↑ E8/9↑ G16↑ Sec F GHr GHi F↑ GHV FV Low
1 2 2 2 1 5 5
2 3 1 2 1 3 1 8 8
3 10 6 4 4 1 1 3 2 21 1 1 22
4 7 2 5 4 1 1 3 1 17 1 1 18
5 14 11 3 3 1 2 4 24 24
6 7 6 1 3 1 2 3 16 16
7 1 1 1 1 5 2 3 13 1 1 14
8 5 4 1 2 2 1 1 11 11
9 7 5 2 1 2 1 1 12 12
10 11 9 2 5 3 3 1 2 25 1 1 26
10 67 46 21 25 9 20 5 20 6 152 3 1 4 156
Two D10 D5 H4 G19 E6/14 H4↑ E8/9 G16 E6/14↑ H4↑↑ E8/9↑ G16↑ Sec F GHr GHi F↑ GHV FV Low
11 9 7 2 1 1 2 2 2 17 1 1 18
12 16 10 6 2 2 2 2 3 27 27
13 9 7 2 2 3 3 1 1 19 1 1 20
14 9 5 4 1 1 1 6 1 19 1 1 2 21
15 5 5 2 1 8 8
16 9 6 3 1 2 1 1 2 1 17 1 1 2 19
17 10 4 6 7 2 2 1 22 2 2 24
18 5 4 1 5 1 2 1 13 13
19 11 9 2 2 4 1 2 20 2 2 4 24
20 10 6 4 5 1 2 1 19 1 1 20
20 160 109 51 53 18 38 13 38 13 333 10 2 4 1 17 350
Two D10 D5 H4 G19 E6/14 H4↑ E8/9 G16 E6/14↑ H4↑↑ E8/9↑ G16↑ Sec F GHr GHi F↑ GHV FV Low
21 12 10 2 1 1 1 2 1 18 18
22 10 5 5 3 1 3 4 1 22 22
23 5 4 1 4 2 1 4 1 17 2 2 19
24 13 9 4 2 1 2 1 1 20 20
25 9 5 4 2 1 1 2 15 1 1 1 3 18
26 5 5 1 1 1 1 1 10 1 1 11
27 5 5 1 1 1 1 1 3 13 1 1 2 15
28 8 7 1 3 1 1 2 1 1 17 2 2 19
29 9 7 2 2 1 1 5 18 1 1 19
30 11 8 3 2 3 2 1 19 2 1 3 22
30 247 174 73 74 25 50 23 57 22 3 1 502 17 4 5 5 31 533
Two D10 D5 H4 G19 E6/14 H4↑ E8/9 G16 E6/14↑ H4↑↑ E8/9↑ G16↑ Sec F GHr GHi F↑ GHV FV Low
31 5 5 3 1 4 1 14 14
32 8 7 1 1 4 3 1 1 18 1 1 19
33 11 9 2 2 1 4 2 2 22 22
34 7 5 2 4 3 1 1 16 16
35 10 4 6 4 1 4 19 2 1 3 22
36 14 11 3 4 1 1 4 1 25 25
37 8 5 3 1 1 1 3 1 15 15
38 14 11 3 4 1 2 1 1 23 23
39 16 12 4 1 1 1 19 1 1 2 21
40 13 11 2 3 1 1 3 21 1 1 2 23
40 353 254 99 101 35 67 30 74 28 4 2 694 21 7 5 6 39 733
Two D10 D5 H4 G19 E6/14 H4↑ E8/9 G16 E6/14↑ H4↑↑ E8/9↑ G16↑ Sec F GHr GHi F↑ GHV FV Low
41 6 6 7 1 3 2 1 1 21 1 1 2 23
42 12 8 4 2 2 1 1 1 19 19
43 5 5 4 2 1 2 2 16 2 1 3 19
44 11 7 4 3 2 5 2 1 1 25 1 1 26
45 11 7 4 8 2 2 3 1 1 28 28
46 12 9 3 4 2 1 4 1 24 1 1 2 26
47 8 8 4 3 2 17 2 2 19
48 6 5 1 3 1 1 2 13 1 1 14
49 10 6 4 3 1 6 3 1 24 3 3 27
50 6 4 2 3 1 2 1 13 1 1 14
50 440 319 121 142 46 89 35 94 40 4 1 3 894 30 10 5 8 1 54 948
Two D10 D5 H4 G19 E6/14 H4↑ E8/9 G16 E6/14↑ H4↑↑ E8/9↑ G16↑ Sec F GHr GHi F↑ GHV FV Low
51 6 3 3 1 1 1 1 11 1 1 1 3 14
52 9 6 3 4 1 1 4 19 1 1 20
53 9 8 1 6 2 2 4 23 1 1 24
54 7 6 1 5 2 1 2 1 18 18
55 7 6 1 1 1 2 4 15 4 4 19
56 10 8 2 2 1 2 1 2 1 19 1 1 20
57 4 2 2 4 5 2 15 15
58 4 4 4 2 1 1 1 13 3 3 16
59 11 5 6 1 1 1 1 1 16 1 1 17
60 14 11 3 4 2 3 2 2 1 28 1 2 3 31
60 521 378 143 173 55 108 40 115 47 5 1 6 1071 42 13 5 9 1 1 71 1142
Two D10 D5 H4 G19 E6/14 H4↑ E8/9 G16 E6/14↑ H4↑↑ E8/9↑ G16↑ Sec F GHr GHi F↑ GHV FV Low
61 9 8 1 2 2 3 1 2 1 20 1 1 21
62 12 6 6 2 4 4 2 1 25 25
63 12 8 4 2 1 3 4 2 1 25 1 1 2 27
64 13 9 4 7 1 2 5 1 29 2 2 31
65 14 5 9 4 1 2 1 1 3 1 27 1 1 28
66 10 6 4 3 2 3 2 20 20
67 13 5 8 2 3 4 2 24 2 1 3 27
68 8 4 4 7 1 1 17 1 1 18
69 9 7 2 3 3 1 1 2 1 20 20
70 17 11 6 3 1 1 3 1 26 26
70 638 447 191 208 71 131 45 138 55 9 2 7 1304 48 13 7 10 2 1 81 1385
Two D10 D5 H4 G19 E6/14 H4↑ E8/9 G16 E6/14↑ H4↑↑ E8/9↑ G16↑ Sec F GHr GHi F↑ GHV FV Low
71 12 7 5 4 2 1 2 2 1 24 24
72 7 5 2 7 1 2 1 3 1 22 1 1 23
73 6 4 2 4 1 3 1 1 16 2 1 3 19
74 4 2 2 3 1 8 1 1 9
75 14 12 2 4 1 1 2 22 1 1 23
76 11 7 4 3 4 1 1 1 21 1 1 2 23
77 10 8 2 3 1 1 1 4 1 21 21
78 13 11 2 5 1 2 1 22 2 2 24
79 12 10 2 1 1 3 1 18 1 1 2 20
80 15 13 2 3 2 2 3 1 26 1 1 2 28
80 742 526 216 245 77 146 54 154 64 12 2 8 1504 56 15 10 11 2 1 95 1599
Two D10 D5 H4 G19 E6/14 H4↑ E8/9 G16 E6/14↑ H4↑↑ E8/9↑ G16↑ Sec F GHr GHi F↑ GHV FV Low
81 7 5 2 2 1 3 1 1 3 18 3 2 5 23
82 13 9 4 2 1 5 1 1 23 1 1 2 25
83 9 7 2 2 3 3 3 20 1 1 2 22
84 5 3 2 1 1 1 4 12 1 1 13
85 11 5 6 1 2 1 1 16 1 1 2 18
86 9 6 3 3 6 1 2 1 1 23 2 2 25
87 11 10 1 1 1 2 1 1 17 1 1 18
88 11 9 2 1 1 3 16 1 1 17
89 11 9 2 4 2 4 1 1 23 1 1 24
90 15 11 4 3 1 3 1 23 1 1 24
90 844 600 244 264 85 167 58 177 71 14 4 10 1 1695 67 17 13 11 4 1 113 1808
Two D10 D5 H4 G19 E6/14 H4↑ E8/9 G16 E6/14↑ H4↑↑ E8/9↑ G16↑ Sec F GHr GHi F↑ GHV FV Low
91 7 6 1 3 3 1 4 1 1 20 2 2 4 24
92 5 4 1 6 2 1 1 2 17 1 1 2 19
93 8 6 2 5 1 1 4 1 20 1 1 21
94 8 6 2 1 2 5 2 1 19 2 2 21
95 15 9 6 1 1 1 2 1 1 22 2 1 3 25
96 14 11 3 6 1 1 1 23 1 1 24
97 11 9 2 6 1 2 2 1 1 24 24
98 5 3 2 1 3 2 1 12 1 1 2 14
99 8 6 2 4 2 1 1 2 1 19 1 1 20
100 11 7 4 3 2 1 17 2 1 3 20
100 936 667 269 297 94 186 63 197 79 15 6 13 2 1888 78 19 15 14 5 1 132 2020
Two D10 D5 H4 G19 E6/14 H4↑ E8/9 G16 E6/14↑ H4↑↑ E8/9↑ G16↑ Sec F GHr GHi F↑ GHV FV Low
N Second Maximal Class Lower Class Tot
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Relative frequency of second 3-class groups,
two-stage towers, and "small" groups


A comparison of the absolute intervals
0 < |d| < 105 and 0 < |d| < 106
shows an increasing frequency of the groups of lower than second maximal class (e ≥ 4)
from 4 / 156 = 2.6% to 132 / 2020 = 6.5%.
Although the groups of second maximal class (e = 3) [Ne] are therefore
decreasing from 152 / 156 = 97.4% to 1888 / 2020 = 93.5%,
their subset of groups with two-stage towers (types D.5 and D.10)
increases from 67 / 156 = 42.9% to 936 / 2020 = 46.3%.

From the computational point of view, the "small" groups with transfer kernel types D.10, D.5, H.4, G.19,
with slightly increasing frequency from 101 / 156 = 64.7% to 1327 / 2020 = 65.7%,
can be identified simply by the ε-invariant and thus considerably faster,
since for all other types the structure of the 3-class group of the
Hilbert 3-class field of K, which is of absolute degree 18, must be determined.

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Relative frequency of simply real cubic fields with fundamental discriminant
and various 3-class numbers


For each of the 2020 complex quadratic fields K with discriminant d
we have four simply real cubic fields L1,…,L4 with fundamental discriminant d.
The total number of these fields is therefore 8080.
By a theorem of F. Gerth III [Ge],
their 3-class groups Cl3(Li) are cyclic and non-trivial.
The dominating part of 7255 / 8080 = 89.8% has 3-class number h3(Li) = 3,
a modest part of 768 / 8080 = 9.5% has 3-class number h3(Li) = 9,
and a negligible part of 57 / 8080 = 0.7% has 3-class number h3(Li) = 27.

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Relative frequency of complex S3-fields
with exotic 3-class groups of type (3,3,3)


For each of the 2020 complex quadratic fields K with discriminant d
we have four complex S3-fields N1,…,N4 with discriminant d3.
The total number of these fields is therefore 8080.
2630 / 8080 = 32.5% of these fields have an exotic 3-class group of type (3,3,3).
Every other field has an almost homogeneous 3-class group of type (3q+r,3q) with 0 ≤ r ≤ 1.


Statistics of second 3-class groups G = Gal(K2|K)
over real quadratic fields K with 3-class group of type (3,3)
and discriminant 0 < d < 107.



K2 denotes the second Hilbert 3-class field of K.
The number of these fields is 2576.

The table gives the number of occurrences of each transfer kernel type (TKT) of G
for relative intervals (N-1)*105 < d < N*105.
"Two" denotes the number of two-stage towers of 3-class fields.
Boldface rows contain data for absolute intervals 0 < d < N*105.

N Maximal Class Second Maximal Class Lower Class Tot
a2/3 a3* a1 a2/3↑ a1↑ Max Two D10 D5 H4 G19 c18 E6/14 H4↑ c21 E8/9 G16 c18↑ c21↑ Sec b.10 d F d* F↑ Low
1 4 1 5 4 5
2 8 2 1 11 10 11
3 10 2 1 13 12 1 1 14
4 5 4 1 10 9 1 1 11
5 10 5 1 16 16 1 1 17
6 10 4 3 17 16 2 1 1 4 21
7 8 4 12 14 1 1 2 14
8 9 5 1 1 16 14 1 1 17
9 10 2 1 13 15 1 2 3 16
10 12 6 2 20 18 1 1 1 3 23
10 86 34 11 2 133 128 5 3 1 2 1 2 1 15 1 1 149
a2/3 a3* a1 a2/3↑ a1↑ Max Two D10 D5 H4 G19 c18 E6/14 H4↑ c21 E8/9 G16 c18↑ c21↑ Sec b.10 d F d* F↑ Low
11 10 4 2 16 16 2 1 1 4 20
12 8 2 2 12 11 1 1 2 14
13 9 3 12 12 1 1 13
14 14 5 4 23 20 1 1 24
15 12 5 1 1 19 17 1 1 20
16 8 2 2 1 13 12 2 1 3 1 1 17
17 15 8 23 26 2 1 1 1 5 28
18 14 8 2 2 26 23 1 1 1 3 29
19 19 6 1 26 27 2 2 28
20 9 5 1 15 16 1 1 1 3 18
20 204 82 24 8 318 308 14 8 3 3 2 1 5 3 1 40 1 1 2 360
a2/3 a3* a1 a2/3↑ a1↑ Max Two D10 D5 H4 G19 c18 E6/14 H4↑ c21 E8/9 G16 c18↑ c21↑ Sec b.10 d F d* F↑ Low
21 14 4 3 1 22 19 1 1 2 24
22 15 3 2 20 20 1 1 2 22
23 10 4 14 14 14
24 10 4 1 15 17 3 1 1 5 1 1 21
25 13 4 2 19 17 19
26 13 7 2 2 24 24 3 1 4 28
27 8 6 1 15 15 1 1 2 17
28 12 9 1 1 23 21 2 2 25
29 17 7 1 1 26 25 1 1 2 28
30 22 4 1 1 1 29 26 29
30 338 134 37 15 1 525 506 23 11 6 4 5 1 5 3 1 59 1 2 3 587
a2/3 a3* a1 a2/3↑ a1↑ Max Two D10 D5 H4 G19 c18 E6/14 H4↑ c21 E8/9 G16 c18↑ c21↑ Sec b.10 d F d* F↑ Low
31 13 15 1 29 29 1 1 2 31
32 16 7 2 25 24 1 1 1 3 28
33 7 6 1 14 16 3 2 1 6 20
34 11 8 2 21 21 2 2 23
35 16 7 23 23 1 1 2 1 1 26
36 18 5 23 23 1 1 24
37 13 6 3 2 24 21 2 1 3 27
38 17 5 2 24 24 1 1 2 26
39 12 4 2 18 17 1 1 2 1 1 21
40 18 8 1 27 28 2 1 1 4 31
40 479 205 51 17 1 753 732 34 14 9 5 9 1 2 8 3 1 86 3 2 5 844
a2/3 a3* a1 a2/3↑ a1↑ Max Two D10 D5 H4 G19 c18 E6/14 H4↑ c21 E8/9 G16 c18↑ c21↑ Sec b.10 d F d* F↑ Low
41 9 6 1 16 16 1 1 1 1 1 5 21
42 22 4 3 29 28 1 1 2 31
43 17 7 3 3 30 26 2 2 32
44 17 9 1 27 28 1 1 2 29
45 13 6 2 21 21 2 1 1 4 1 1 26
46 14 9 2 1 26 23 1 1 Gap 27
47 8 5 2 15 13 1 1 16
48 14 7 3 24 24 2 1 1 1 5 29
49 17 4 21 22 1 1 2 23
50 14 5 1 20 20 1 1 2 22
50 624 267 68 22 1 982 953 43 19 13 7 11 1 3 9 5 1 112 4 2 6 1100
a2/3 a3* a1 a2/3↑ a1↑ Max Two D10 D5 H4 G19 c18 E6/14 H4↑ c21 E8/9 G16 c18↑ c21↑ Sec b.10 d F d* F↑ Low
51 7 9 2 2 20 20 3 1 1 5 Gap 25
52 16 8 3 27 24 1 1 28
53 13 4 1 1 19 17 1 1 2 21
54 22 8 4 2 36 33 3 1 4 40
55 11 5 2 18 16 1 1 19
56 13 8 1 22 21 1 2 3 25
57 14 7 1 22 22 1 1 23
58 16 8 24 25 1 1 2 26
59 10 7 2 19 19 2 1 3 22
60 16 7 2 25 25 2 1 1 4 29
60 762 338 84 29 1 1214 1175 55 20 15 8 16 2 3 12 6 1 138 4 2 6 1358
a2/3 a3* a1 a2/3↑ a1↑ Max Two D10 D5 H4 G19 c18 E6/14 H4↑ c21 E8/9 G16 c18↑ c21↑ Sec b.10 d F d* F↑ Low
61 15 4 1 20 23 2 2 3 1 8 Gap 28
62 12 6 2 1 21 20 1 1 1 3 24
63 21 9 1 31 32 2 1 1 4 35
64 13 8 1 2 24 22 1 1 1 3 27
65 15 9 1 2 27 24 2 1 3 30
66 16 9 2 27 26 1 1 1 3 30
67 12 12 2 26 26 1 1 1 3 29
68 15 8 1 1 25 25 2 1 1 4 29
69 13 8 2 23 22 1 1 2 25
70 10 5 3 18 15 1 1 19
70 904 416 96 39 1 1456 1410 65 25 21 9 18 3 3 19 8 1 172 4 2 6 1634
a2/3 a3* a1 a2/3↑ a1↑ Max Two D10 D5 H4 G19 c18 E6/14 H4↑ c21 E8/9 G16 c18↑ c21↑ Sec b.10 d F d* F↑ Low
71 19 7 3 2 31 26 Gap 31
72 15 10 25 25 1 1 1 1 1 5 30
73 12 6 1 1 20 20 1 1 2 22
74 11 11 2 24 25 3 1 4 28
75 13 13 3 2 31 27 1 1 32
76 16 11 1 2 30 30 2 1 1 1 5 35
77 23 8 4 1 36 33 2 2 38
78 18 13 2 33 33 1 1 1 3 6 39
79 12 10 1 1 24 24 2 1 3 27
80 16 7 1 24 23 1 1 25
80 1059 512 114 48 1 1734 1676 74 31 23 11 24 4 3 20 10 1 201 4 2 6 1941
a2/3 a3* a1 a2/3↑ a1↑ Max Two D10 D5 H4 G19 c18 E6/14 H4↑ c21 E8/9 G16 c18↑ c21↑ Sec b.10 d F d* F↑ Low
81 11 5 1 1 18 17 1 2 1 4 Gap 22
82 16 13 1 1 31 31 1 1 2 2 1 3 36
83 13 6 3 22 20 1 1 2 4 26
84 25 15 2 1 43 42 1 1 1 3 1 1 47
85 13 11 1 25 25 1 1 1 1 27
86 14 7 2 3 26 22 1 1 27
87 19 6 3 1 29 27 1 1 1 3 32
88 18 7 3 1 29 28 2 1 1 1 5 34
89 23 9 1 2 35 34 1 1 1 1 4 39
90 14 13 2 2 31 29 1 1 2 33
90 1225 604 132 61 1 2023 1951 84 38 26 11 27 5 3 22 12 1 1 230 6 2 1 1 1 11 2264
a2/3 a3* a1 a2/3↑ a1↑ Max Two D10 D5 H4 G19 c18 E6/14 H4↑ c21 E8/9 G16 c18↑ c21↑ Sec b.10 d F d* F↑ Low
91 21 7 3 1 32 28 1 1 33
92 14 13 3 1 31 30 2 1 1 1 5 36
93 16 11 27 27 27
94 18 8 3 29 30 1 3 4 33
95 13 9 1 23 24 1 1 1 1 1 5 28
96 24 11 1 3 39 35 1 1 40
97 17 6 2 2 27 26 2 1 1 4 31
98 6 10 3 19 18 1 1 2 21
99 12 9 2 23 24 2 1 3 1 1 27
100 20 9 1 30 30 1 1 1 1 1 5 1 1 36
100 1386 697 147 72 1 2303 2223 93 47 27 11 29 7 3 25 14 2 2 260 8 2 1 1 1 13 2576
a2/3 a3* a1 a2/3↑ a1↑ Max Two D10 D5 H4 G19 c18 E6/14 H4↑ c21 E8/9 G16 c18↑ c21↑ Sec b.10 d F d* F↑ Low
N Maximal Class Second Maximal Class Lower Class Tot

Second 3-class groups G = Gal(K2|K)
of lower than second maximal class
show an astonishing and remarkable gap of length greater than 3.6 millions
for discriminants 4.5*106 < d < 8.1*106,
followed by a real explosion between 8.1*106 and 8.5*106,
where the transfer kernel types F.13↑, F.13, and d.25* appear for the first time.

The latest of all occurrences is revealed by transfer kernel type G.16 at d = 8711453.

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Relative frequency of second 3-class groups
and two-stage towers


A comparison of the absolute intervals
0 < d < 106 and 0 < d < 107
shows that all frequencies remain rather stable.
Groups of maximal class (e = 2) [Bl], [Mi] are clearly dominating
with 133 / 149 = 89% and 2303 / 2576 = 89.4%,
the contribution of groups of second maximal class (e = 3) [Ne] remains modest
with 15 / 149 = 10% and 260 / 2576 = 10.1%,
and the groups of lower than second maximal class (e ≥ 4) are almost negligible
with 1 / 149 = 0.7% and 13 / 2576 = 0.5%.
The broad subset of groups with two-stage towers consists of
groups of maximal class (type a.3* and the ground states of types a.2 and a.3)
and groups of second maximal class (types D.5 and D.10)
with 128 / 149 = 86% and 2223 / 2576 = 86.3%.

From the computational point of view, all groups of maximal class
and the groups with transfer kernel types D.10, D.5, H.4, G.19, c.18, c.21,
with constant frequency of 147 / 149 = 98.7% and 2537 / 2576 = 98.5%,
can be identified already by the structure of the 3-class groups
of the four unramified cyclic cubic extension fields N1,…,N4 of K
and thus considerably faster,
since for the other types the structure of the 3-class group of the
Hilbert 3-class field of K, which is of absolute degree 18, must be determined.

The value 98.5% in comparison to 65.7% for complex fields
means that the classification of real fields is much less time consuming.

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Relative frequency of totally real cubic fields with fundamental discriminant
and various 3-class numbers


For each of the 2576 real quadratic fields K with discriminant d
we have four totally real cubic fields L1,…,L4 with fundamental discriminant d.
The total number of these fields is therefore 10304.
By a theorem of F. Gerth III [Ge],
their 3-class groups Cl3(Li) are cyclic and non-trivial.
The clearly dominating part of 9976 / 10304 = 96.8% has 3-class number h3(Li) = 3,
a very modest part of 323 / 10304 = 3.1% has 3-class number h3(Li) = 9,
and a negligible part of 5 / 10304 = 0.1% has 3-class number h3(Li) = 27.

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Relative frequency of totally real S3-fields
with exotic 3-class groups of type (3,3,3)


For each of the 2576 real quadratic fields K with discriminant d
we have four totally real S3-fields N1,…,N4 with discriminant d3.
The total number of these fields is therefore 10304.
1030 / 10304 = 10.0% of these fields have an exotic 3-class group of type (3,3,3).
Every other field has an almost homogeneous 3-class group of type (3q+r,3q) with 0 ≤ r ≤ 1.


References:

[Bl] Norman Blackburn,
On a special class of p-groups,
Acta Math. 100 (1958), 45 - 92.

[Ge] Frank Gerth III,
Ranks of 3-class groups of non-Galois cubic fields,
Acta Arith. 30 (1976), 307 - 322.

[Mi] R. J. Miech,
Metabelian p-groups of maximal class,
Trans. Amer. Math. Soc. 152 (1970), 331 - 373.

[Ne] Brigitte Nebelung,
Klassifikation metabelscher 3-Gruppen
mit Faktorkommutatorgruppe vom Typ (3,3)
und Anwendung auf das Kapitulationsproblem,
Inauguraldissertation, Univ. zu Köln, 1989.

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