Properties

Label 12T103
Order \(192\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $103$
CHM label :  $1/2[E(4)^{3}]S(3)$
Parity:  $1$
Primitive:  No
Generators:   ( 1, 3, 7, 9)( 2, 5)( 4, 6,10,12)( 8,11), ( 1, 5, 9,10, 2, 6)( 3, 4, 8,12, 7,11), ( 1, 7)( 2,12, 5, 3)( 4,10)( 6,11, 9, 8)
$|\Aut(F/K)|$:  $4$
Low degree resolvents:  
2: 2T1, 2T1, 2T1
4: 4T2
6: 3T2
12: 6T3
24: 4T5, 4T5, 4T5
48: 6T11, 6T11, 6T11
96: 8T34

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$, $S_4\times C_2$, $S_4\times C_2$

Low degree siblings

12T100a, 12T100b, 12T100c, 12T101a, 12T101b, 12T101c, 12T101d, 12T101e, 12T101f, 12T103b, 12T103c, 12T103d, 12T103e, 12T103f, 12T106, 16T429
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 $ $3$ $2$ $( 3,12)( 6, 9)$
$ 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 $ $3$ $2$ $( 2, 5)( 3,12)( 6, 9)( 8,11)$
$ 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 $ $12$ $2$ $( 1, 2)( 3, 9)( 4,11)( 5,10)( 6,12)( 7, 8)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 2, 3)( 4,11, 6)( 5,12,10)( 7, 8, 9)$
$ 6, 6 $ $32$ $6$ $( 1, 2, 3,10, 5,12)( 4,11, 6, 7, 8, 9)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2, 4,11)( 3,12)( 5, 7, 8,10)( 6, 9)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2, 7, 8)( 3,12)( 4,11,10, 5)( 6, 9)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2,10, 5)( 3, 6)( 4,11, 7, 8)( 9,12)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2,10, 5)( 3, 9)( 4,11, 7, 8)( 6,12)$
$ 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 $ $3$ $2$ $( 1, 7)( 2, 5)( 3, 9)( 4,10)( 6,12)( 8,11)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 5)( 3,12)( 4, 7)( 6, 9)( 8,11)$

Group invariants

Order:  $192=2^{6} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [192, 1538]
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 4a 4b 2b 2c 2d 2e 2f 2g 3a 6a 4c 4d 4e 4f 2h 2i 2j 2k
     2P 1a 1a 2d 2e 1a 1a 1a 1a 1a 1a 3a 3a 2e 2d 2b 2b 1a 1a 1a 1a
     3P 1a 2a 4a 4b 2b 2c 2d 2e 2f 2g 1a 2k 4c 4d 4e 4f 2h 2i 2j 2k
     5P 1a 2a 4a 4b 2b 2c 2d 2e 2f 2g 3a 6a 4c 4d 4e 4f 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      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  3  1 -1 -1 -1  1  .  . -1  1  1 -1  1 -1  1 -3
X.8      3 -3  1 -1  3  1 -1 -1  1 -1  .  .  1 -1 -1  1  1 -1  1 -3
X.9      3 -1 -1  1 -1 -1  3 -1 -1 -1  .  .  1 -1  1  1  3 -1 -1  3
X.10     3 -1  1 -1 -1 -1  3 -1  1  1  .  . -1  1 -1 -1  3 -1 -1  3
X.11     3  1 -1 -1 -1  1  3 -1  1 -1  .  .  1  1  1 -1 -3 -1  1 -3
X.12     3  1  1  1 -1  1  3 -1 -1  1  .  . -1 -1 -1  1 -3 -1  1 -3
X.13     3  3 -1 -1  3 -1 -1 -1  1  1  .  . -1 -1  1  1 -1 -1 -1  3
X.14     3  3  1  1  3 -1 -1 -1 -1 -1  .  .  1  1 -1 -1 -1 -1 -1  3
X.15     3 -1 -1  1 -1 -1 -1  3  1  1  .  .  1 -1 -1 -1 -1 -1  3  3
X.16     3 -1  1 -1 -1 -1 -1  3 -1 -1  .  . -1  1  1  1 -1 -1  3  3
X.17     3  1 -1 -1 -1  1 -1  3 -1  1  .  .  1  1 -1  1  1 -1 -3 -3
X.18     3  1  1  1 -1  1 -1  3  1 -1  .  . -1 -1  1 -1  1 -1 -3 -3
X.19     6  2  .  . -2 -2 -2 -2  .  .  .  .  .  .  .  .  2  2  2 -6
X.20     6 -2  .  . -2  2 -2 -2  .  .  .  .  .  .  .  . -2  2 -2  6