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Properties

Label 12T98
Order \(192\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2.A_4:C_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $98$
Group :  $C_2^2.A_4:C_4$
CHM label :  $1/2[4^{3}]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,9)(6,12), (1,8,7,2)(3,6,9,12)(4,11,10,5), (1,10,7,4)(3,6,9,12), (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:  $C_2$
4:  $C_4$
6:  $S_3$
12:  $C_3 : C_4$
24:  $S_4$
48:  12T27
96:  12T62

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$

Low degree siblings

12T98, 24T317, 24T481, 24T482 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, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3, 9)( 6,12)$
$ 8, 1, 1, 1, 1 $ $12$ $8$ $( 2, 3, 5, 6, 8, 9,11,12)$
$ 8, 1, 1, 1, 1 $ $12$ $8$ $( 2, 3,11,12, 8, 9, 5, 6)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2, 5, 8,11)( 3, 6, 9,12)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 2, 5, 8,11)( 3,12, 9, 6)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 8)( 3, 9)( 5,11)( 6,12)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2,11, 8, 5)( 3,12, 9, 6)$
$ 4, 2, 2, 2, 2 $ $12$ $4$ $( 1, 2)( 3, 6, 9,12)( 4, 5)( 7, 8)(10,11)$
$ 4, 2, 2, 2, 2 $ $12$ $4$ $( 1, 2)( 3,12, 9, 6)( 4, 5)( 7, 8)(10,11)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)$
$ 6, 6 $ $32$ $6$ $( 1, 2, 3, 7, 8, 9)( 4, 5, 6,10,11,12)$
$ 8, 2, 2 $ $12$ $8$ $( 1, 2, 4, 5, 7, 8,10,11)( 3, 9)( 6,12)$
$ 4, 4, 4 $ $12$ $4$ $( 1, 2, 7, 8)( 3, 6, 9,12)( 4, 5,10,11)$
$ 4, 4, 4 $ $12$ $4$ $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$
$ 8, 2, 2 $ $12$ $8$ $( 1, 2,10,11, 7, 8, 4, 5)( 3, 9)( 6,12)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 9)( 6,12)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 4, 7,10)( 2, 8)( 3,12, 9, 6)( 5,11)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 7)( 2,11, 8, 5)( 3,12, 9, 6)( 4,10)$

Group invariants

Order:  $192=2^{6} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [192, 182]
Character table:    
      2  6  6  4  4   6  5  6   6  4  4  1  1  4  4  4  4  6  5  6  6
      3  1  .  .  .   .  .  .   .  .  .  1  1  .  .  .  .  .  .  1  .

        1a 2a 8a 8b  4a 4b 2b  4c 4d 4e 3a 6a 8c 4f 4g 8d 4h 4i 2c 4j
     2P 1a 1a 4a 4c  2b 2b 1a  2b 2a 2a 3a 3a 4a 2c 2c 4c 2b 2b 1a 2b
     3P 1a 2a 8b 8a  4c 4b 2b  4a 4e 4d 1a 2c 8d 4g 4f 8c 4j 4i 2c 4h
     5P 1a 2a 8a 8b  4a 4b 2b  4c 4d 4e 3a 6a 8c 4f 4g 8d 4h 4i 2c 4j
     7P 1a 2a 8b 8a  4c 4b 2b  4a 4e 4d 3a 6a 8d 4g 4f 8c 4j 4i 2c 4h

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

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