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Properties

Label 12T48
Order \(96\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times S_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $48$
Group :  $C_2^2\times S_4$
CHM label :  $2S_{4}(6)[x]2=[1/4.2^{6}]S(3)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7)(6,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11), (2,10)(3,11)(4,8)(5,9)
$|\Aut(F/K)|$:  $4$

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:  $S_4$, $S_3 \times C_2^2$
48:  $S_4\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$, $S_4\times C_2$ x 2

Low degree siblings

12T48 x 11, 16T182 x 4, 24T125, 24T126 x 6, 24T150 x 3, 24T151 x 4, 24T152 x 4, 32T388

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

Group invariants

Order:  $96=2^{5} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [96, 226]
Character table:    
      2  5  5  4  4  5  4  4  2  2  4  4  2  2  4  4  5  5  5  5  5
      3  1  .  .  .  .  .  .  1  1  .  .  1  1  .  .  .  .  1  1  1

        1a 2a 2b 4a 2c 2d 2e 6a 6b 4b 4c 6c 3a 4d 2f 2g 2h 2i 2j 2k
     2P 1a 1a 1a 2c 1a 1a 1a 3a 3a 2c 2c 3a 3a 2c 1a 1a 1a 1a 1a 1a
     3P 1a 2a 2b 4a 2c 2d 2e 2k 2i 4b 4c 2j 1a 4d 2f 2g 2h 2i 2j 2k
     5P 1a 2a 2b 4a 2c 2d 2e 6a 6b 4b 4c 6c 3a 4d 2f 2g 2h 2i 2j 2k

X.1      1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
X.2      1 -1 -1  1  1 -1  1  1 -1  1 -1 -1  1 -1  1 -1  1 -1 -1  1
X.3      1 -1 -1  1  1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1
X.4      1 -1  1 -1  1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1  1 -1 -1
X.5      1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1 -1  1 -1 -1  1
X.6      1  1 -1 -1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1  1  1  1
X.7      1  1 -1 -1  1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1 -1  1 -1
X.8      1  1  1  1  1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1  1 -1
X.9      2 -2  .  .  2  .  . -1  1  .  .  1 -1  .  . -2  2 -2 -2  2
X.10     2 -2  .  .  2  .  .  1 -1  .  .  1 -1  .  .  2 -2  2 -2 -2
X.11     2  2  .  .  2  .  . -1 -1  .  . -1 -1  .  .  2  2  2  2  2
X.12     2  2  .  .  2  .  .  1  1  .  . -1 -1  .  . -2 -2 -2  2 -2
X.13     3 -1 -1  1 -1 -1 -1  .  .  1  1  .  .  1 -1 -1 -1  3  3  3
X.14     3 -1 -1  1 -1  1  1  .  . -1 -1  .  .  1 -1  1  1 -3  3 -3
X.15     3 -1  1 -1 -1 -1 -1  .  .  1  1  .  . -1  1  1  1 -3  3 -3
X.16     3 -1  1 -1 -1  1  1  .  . -1 -1  .  . -1  1 -1 -1  3  3  3
X.17     3  1 -1 -1 -1 -1  1  .  . -1  1  .  .  1  1  1 -1 -3 -3  3
X.18     3  1 -1 -1 -1  1 -1  .  .  1 -1  .  .  1  1 -1  1  3 -3 -3
X.19     3  1  1  1 -1 -1  1  .  . -1  1  .  . -1 -1 -1  1  3 -3 -3
X.20     3  1  1  1 -1  1 -1  .  .  1 -1  .  . -1 -1  1 -1 -3 -3  3