Galois Group: 12T100

• Label: 12T100
• Order: \(192\)
• n: \(12\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No
• Group: $C_2\times C_2^2:S_4$

Group action invariants

Degree $n$ :		$12$
Transitive number $t$ :		$100$
Group :		$C_2\times C_2^2:S_4$
CHM label :		$[(1/2.2^{2})^{3}]S_{4}(6d)_{2}$
Parity:		$-1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,12)(2,3), (2,3)(4,8)(5,9)(6,11)(7,10), (1,7,11)(2,4,8)(3,5,9)(6,10,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:		$2$

Low degree resolvents

|G/N|	Galois groups for stem field(s)
2:	$C_2$ x 3
4:	$C_2^2$
6:	$S_3$
12:	$D_{6}$
24:	$S_4$ x 3
48:	$S_4\times C_2$ x 3
96:	$V_4^2:S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$

Low degree siblings

12T100 x 2, 12T101 x 6, 12T103 x 6, 12T106, 16T429, 24T432 x 3, 24T485 x 3, 24T486 x 6, 24T487 x 6, 24T488 x 3, 24T489 x 3, 24T490 x 3, 24T491 x 2, 24T492 x 6, 24T493 x 6, 24T508 x 3, 24T509 x 6, 24T510, 24T511, 32T2212 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle Type	Size	Order	Representative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $	$3$	$2$	$( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $	$3$	$2$	$( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $	$6$	$2$	$( 4, 6)( 5, 7)( 8,10)( 9,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $	$3$	$2$	$( 4, 6)( 5, 7)( 8,11)( 9,10)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $	$3$	$2$	$( 4, 7)( 5, 6)( 8,10)( 9,11)$
$ 2, 2, 2, 2, 2, 1, 1 $	$12$	$2$	$( 2, 3)( 4, 8)( 5, 9)( 6,11)( 7,10)$
$ 4, 4, 2, 1, 1 $	$12$	$4$	$( 2, 3)( 4, 8, 5, 9)( 6,11, 7,10)$
$ 4, 4, 2, 1, 1 $	$12$	$4$	$( 2, 3)( 4,10, 5,11)( 6, 9, 7, 8)$
$ 2, 2, 2, 2, 2, 1, 1 $	$12$	$2$	$( 2, 3)( 4,10)( 5,11)( 6, 9)( 7, 8)$
$ 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1, 2)( 3,12)( 4, 5)( 6, 7)( 8,10)( 9,11)$
$ 2, 2, 2, 2, 2, 2 $	$6$	$2$	$( 1, 2)( 3,12)( 4, 5)( 6, 7)( 8,11)( 9,10)$
$ 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1, 2)( 3,12)( 4, 6)( 5, 7)( 8, 9)(10,11)$
$ 4, 4, 4 $	$12$	$4$	$( 1, 2,12, 3)( 4, 8, 6,11)( 5, 9, 7,10)$
$ 4, 4, 4 $	$12$	$4$	$( 1, 2,12, 3)( 4, 8, 7,10)( 5, 9, 6,11)$
$ 4, 4, 4 $	$12$	$4$	$( 1, 2,12, 3)( 4,10, 7, 8)( 5,11, 6, 9)$
$ 4, 4, 4 $	$12$	$4$	$( 1, 2,12, 3)( 4,10, 6, 9)( 5,11, 7, 8)$
$ 3, 3, 3, 3 $	$32$	$3$	$( 1, 4, 8)( 2, 7,11)( 3, 6,10)( 5, 9,12)$
$ 6, 6 $	$32$	$6$	$( 1, 4, 8,12, 5, 9)( 2, 7,11, 3, 6,10)$
$ 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$

Group invariants

Order:		$192=2^{6} \cdot 3$
Cyclic:		No
Abelian:		No
Solvable:		Yes
GAP id:		[192, 1538]

Character table:

2 6 6 6 5 6 6 4 4 4 4 6 5 6 4 4 4 4 1 1 6 3 1 . . . . . . . . . . . . . . . . 1 1 1 1a 2a 2b 2c 2d 2e 2f 4a 4b 2g 2h 2i 2j 4c 4d 4e 4f 3a 6a 2k 2P 1a 1a 1a 1a 1a 1a 1a 2b 2b 1a 1a 1a 1a 2h 2j 2j 2h 3a 3a 1a 3P 1a 2a 2b 2c 2d 2e 2f 4a 4b 2g 2h 2i 2j 4c 4d 4e 4f 1a 2k 2k 5P 1a 2a 2b 2c 2d 2e 2f 4a 4b 2g 2h 2i 2j 4c 4d 4e 4f 3a 6a 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 . . . . 2 -2 2 . . . . -1 1 -2 X.6 2 2 2 2 2 2 . . . . 2 2 2 . . . . -1 -1 2 X.7 3 -3 3 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 . . -3 X.8 3 -3 3 -1 1 1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 . . -3 X.9 3 -1 -1 -1 3 -1 -1 1 1 -1 -1 -1 3 1 -1 -1 1 . . 3 X.10 3 -1 -1 -1 3 -1 1 -1 -1 1 -1 -1 3 -1 1 1 -1 . . 3 X.11 3 1 -1 -1 -3 1 -1 -1 1 1 -1 1 3 -1 -1 1 1 . . -3 X.12 3 1 -1 -1 -3 1 1 1 -1 -1 -1 1 3 1 1 -1 -1 . . -3 X.13 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 . . 3 X.14 3 3 3 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 . . 3 X.15 3 -1 -1 -1 -1 3 -1 1 1 -1 3 -1 -1 -1 1 1 -1 . . 3 X.16 3 -1 -1 -1 -1 3 1 -1 -1 1 3 -1 -1 1 -1 -1 1 . . 3 X.17 3 1 -1 -1 1 -3 -1 -1 1 1 3 1 -1 1 1 -1 -1 . . -3 X.18 3 1 -1 -1 1 -3 1 1 -1 -1 3 1 -1 -1 -1 1 1 . . -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