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Properties

Label 12T134
Order \(384\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^5.(C_2\times C_6)$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $134$
Group :  $C_2^5.(C_2\times C_6)$
CHM label :  $[2^{6}]6=2wr6$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12), (1,3,5,7,9,11)(2,4,6,8,10,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $C_6$ x 3
8:  $D_{4}$
12:  $A_4$, $C_6\times C_2$
24:  $A_4\times C_2$ x 3, $D_4 \times C_3$
48:  $C_2^2 \times A_4$
96:  $C_2^4:C_6$, 12T51
192:  12T87

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: None

Degree 6: $C_6$

Low degree siblings

12T134 x 3, 12T142 x 4, 16T719 x 4, 24T1156 x 4, 24T1157 x 4, 24T1158 x 8, 24T1159 x 4, 24T1160 x 8, 24T1161 x 8, 24T1162 x 8, 24T1163 x 4, 24T1164 x 4, 24T1165 x 4, 24T1166 x 4, 24T1167 x 4, 24T1168 x 4, 24T1169 x 4, 24T1170 x 4, 24T1171 x 2, 24T1172 x 2, 24T1220 x 4, 24T1221 x 4, 24T1222 x 2, 24T1223 x 2, 32T9328 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 8, 9)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 6, 7)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 4, 5)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 4, 5)( 6, 7)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 2, 3)( 6, 7)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 2, 3)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 1, 1 $ $6$ $2$ $( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 6, 6 $ $32$ $6$ $( 1, 2, 4, 6, 8,10)( 3, 5, 7, 9,11,12)$
$ 12 $ $32$ $12$ $( 1, 2, 4, 6, 8,10,12, 3, 5, 7, 9,11)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 4, 8)( 2, 6,10)( 3, 7,11)( 5, 9,12)$
$ 6, 3, 3 $ $32$ $6$ $( 1, 4, 8)( 2, 6,10, 3, 7,11)( 5, 9,12)$
$ 6, 6 $ $16$ $6$ $( 1, 4, 8,12, 5, 9)( 2, 6,10, 3, 7,11)$
$ 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 6)( 2, 8)( 3, 9)( 4,10)( 5,11)( 7,12)$
$ 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 6)( 2, 8)( 3, 9)( 4,10, 5,11)( 7,12)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 6)( 2, 8, 3, 9)( 4,10, 5,11)( 7,12)$
$ 4, 4, 4 $ $8$ $4$ $( 1, 6,12, 7)( 2, 8, 3, 9)( 4,10, 5,11)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 8, 4)( 2,10, 6)( 3,11, 7)( 5,12, 9)$
$ 6, 3, 3 $ $32$ $6$ $( 1, 8, 4)( 2,10, 7, 3,11, 6)( 5,12, 9)$
$ 6, 6 $ $16$ $6$ $( 1, 8, 5,12, 9, 4)( 2,10, 7, 3,11, 6)$
$ 6, 6 $ $32$ $6$ $( 1,10, 8, 6, 4, 2)( 3,12,11, 9, 7, 5)$
$ 12 $ $32$ $12$ $( 1,10, 9, 7, 5, 3,12,11, 8, 6, 4, 2)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$

Group invariants

Order:  $384=2^{7} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [384, 5790]
Character table: Data not available.