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Properties

Label 12T138
Order \(384\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_4^2:C_3:C_2^2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $138$
Group :  $C_2\times C_4^2:C_3:C_2^2$
CHM label :  $[(1/2.2^{2})^{3}]2S_{4}(6)_{4}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(2,3), (1,11,5,3,9,7)(2,8,6,12,10,4), (4,10)(5,11)(6,9)(7,8), (4,8)(5,9)(6,10)(7,11)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$
8:  $C_2^3$
12:  $D_{6}$ x 3
24:  $S_4$, $S_3 \times C_2^2$
48:  $S_4\times C_2$ x 3
96:  12T48
192:  12T112

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4\times C_2$

Low degree siblings

12T138 x 3, 12T140 x 4, 24T1199 x 2, 24T1200 x 4, 24T1201 x 2, 24T1202 x 8, 24T1203 x 8, 24T1207 x 2, 24T1208 x 4, 24T1209 x 2, 24T1210 x 4, 24T1211 x 8, 24T1212 x 8, 24T1213 x 8, 24T1214 x 8, 32T9440 x 4

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, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 4, 6, 5, 7)( 8,10, 9,11)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 4, 6, 5, 7)( 8,11, 9,10)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 4, 8)( 5, 9)( 6,10)( 7,11)$
$ 4, 4, 1, 1, 1, 1 $ $12$ $4$ $( 4, 8, 5, 9)( 6,10, 7,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 2, 3)( 6, 7)( 8,10)( 9,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 2, 3)( 6, 7)( 8,11)( 9,10)$
$ 8, 2, 1, 1 $ $24$ $8$ $( 2, 3)( 4, 8, 6,11, 5, 9, 7,10)$
$ 8, 2, 1, 1 $ $24$ $8$ $( 2, 3)( 4, 8, 7,10, 5, 9, 6,11)$
$ 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3,12)( 4, 6)( 5, 7)( 8,10)( 9,11)$
$ 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3,12)( 4, 6)( 5, 7)( 8,11)( 9,10)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3,12)( 4, 8)( 5, 9)( 6,11)( 7,10)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2)( 3,12)( 4, 8, 5, 9)( 6,11, 7,10)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3,12)( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2)( 3,12)( 4,10, 5,11)( 6, 8, 7, 9)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 2,12, 3)( 4, 5)( 6, 7)( 8,10, 9,11)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 2,12, 3)( 4, 5)( 6, 7)( 8,11, 9,10)$
$ 8, 4 $ $24$ $8$ $( 1, 2,12, 3)( 4, 8, 6,10, 5, 9, 7,11)$
$ 8, 4 $ $24$ $8$ $( 1, 2,12, 3)( 4, 8, 7,11, 5, 9, 6,10)$
$ 6, 6 $ $32$ $6$ $( 1, 4, 8, 2, 6,11)( 3, 7,10,12, 5, 9)$
$ 6, 6 $ $32$ $6$ $( 1, 4, 8, 3, 7,10)( 2, 6,11,12, 5, 9)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 4)( 2, 7)( 3, 6)( 5,12)( 8, 9)(10,11)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 4, 8)( 2, 7,11)( 3, 6,10)( 5, 9,12)$
$ 6, 6 $ $32$ $6$ $( 1, 4, 8,12, 5, 9)( 2, 7,11, 3, 6,10)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 4,12, 5)( 2, 7, 3, 6)( 8, 9)(10,11)$
$ 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, 17949]
Character table: Data not available.