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Properties

Label 12T139
Order \(384\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times C_2^2:S_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $139$
Group :  $C_2^2\times C_2^2:S_4$
CHM label :  $[E(4)^{3}]S(3)=E(4)wrS(3)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,12)(6,9), (3,9)(6,12), (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $4$

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:  $S_4$ x 3, $S_3 \times C_2^2$
48:  $S_4\times C_2$ x 9
96:  $V_4^2:S_3$, 12T48 x 3
192:  12T100 x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4\times C_2$ x 3

Low degree siblings

12T139 x 23, 24T1114 x 36, 24T1115 x 18, 24T1124 x 2, 24T1125 x 9, 24T1126 x 18, 24T1130 x 3, 24T1204 x 6, 24T1205 x 36, 24T1206 x 36, 32T9428

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

Group invariants

Order:  $384=2^{7} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [384, 20163]
Character table: Data not available.