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Properties

Label 12T53
Order \(96\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4\times S_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $53$
Group :  $C_4\times S_4$
CHM label :  $[1/2.4^{2}]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7)(3,9)(4,10)(6,12), (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$
8:  $C_4\times C_2$
12:  $D_{6}$
24:  $S_4$, $S_3 \times C_4$
48:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$

Low degree siblings

12T53, 16T181 x 2, 24T129, 24T130, 24T167, 24T168 x 2, 24T169 x 2, 32T387

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

Group invariants

Order:  $96=2^{5} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [96, 186]
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 4a 2b 2c 4b 4c 12a 12b 4d 4e 6a 3a 2d 4f 4g 4h 4i 2e 4j
     2P 1a 1a 2c 1a 1a 2a 2a  6a  6a 2e 2e 3a 3a 1a 2c 2e 2e 2e 1a 2e
     3P 1a 2a 4a 2b 2c 4c 4b  4g  4j 4e 4d 2e 1a 2d 4f 4j 4i 4h 2e 4g
     5P 1a 2a 4a 2b 2c 4b 4c 12a 12b 4d 4e 6a 3a 2d 4f 4g 4h 4i 2e 4j
     7P 1a 2a 4a 2b 2c 4c 4b 12b 12a 4e 4d 6a 3a 2d 4f 4j 4i 4h 2e 4g
    11P 1a 2a 4a 2b 2c 4c 4b 12b 12a 4e 4d 6a 3a 2d 4f 4j 4i 4h 2e 4g

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  A -A   A  -A -A  A -1  1 -1  1 -A  A -A -1  A
X.6      1 -1 -1  1  1 -A  A  -A   A  A -A -1  1 -1  1  A -A  A -1 -A
X.7      1 -1  1 -1  1  A -A  -A   A -A  A -1  1  1 -1  A -A  A -1 -A
X.8      1 -1  1 -1  1 -A  A   A  -A  A -A -1  1  1 -1 -A  A -A -1  A
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  .  .   A  -A  .  .  1 -1  .  .  B -B  B -2 -B
X.12     2 -2  .  .  2  .  .  -A   A  .  .  1 -1  .  . -B  B -B -2  B
X.13     3 -1 -1  1 -1 -1 -1   .   .  1  1  .  .  1 -1  3 -1 -1  3  3
X.14     3 -1 -1  1 -1  1  1   .   . -1 -1  .  .  1 -1 -3  1  1  3 -3
X.15     3 -1  1 -1 -1 -1 -1   .   .  1  1  .  . -1  1 -3  1  1  3 -3
X.16     3 -1  1 -1 -1  1  1   .   . -1 -1  .  . -1  1  3 -1 -1  3  3
X.17     3  1 -1 -1 -1  A -A   .   .  A -A  .  .  1  1  C -A  A -3 -C
X.18     3  1 -1 -1 -1 -A  A   .   . -A  A  .  .  1  1 -C  A -A -3  C
X.19     3  1  1  1 -1  A -A   .   .  A -A  .  . -1 -1 -C  A -A -3  C
X.20     3  1  1  1 -1 -A  A   .   . -A  A  .  . -1 -1  C -A  A -3 -C

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i
C = 3*E(4)
  = 3*Sqrt(-1) = 3i