Galois Group: 12T97

• Label: 12T97
• Order: \(192\)
• n: \(12\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No
• Group: $C_2\times C_4^2:C_3:C_2$

Group action invariants

Degree $n$ :		$12$
Transitive number $t$ :		$97$
Group :		$C_2\times C_4^2:C_3:C_2$
CHM label :		$[(1/2.2^{2})^{3}]S_{4}(6c)_{4}$
Parity:		$1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,12)(2,3), (1,3)(2,12)(4,8)(5,9)(6,11)(7,10), (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:	$S_4$
48:	$S_4\times C_2$
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

12T95 x 2, 12T96, 24T471, 24T472 x 2, 24T473 x 2, 24T474 x 2, 24T475 x 2, 24T476, 24T477, 24T478, 24T479 x 2, 24T480 x 2, 32T2216 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 $	$3$	$2$	$( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 4, 4, 1, 1, 1, 1 $	$6$	$4$	$( 4, 6, 5, 7)( 8,10, 9,11)$
$ 4, 4, 1, 1, 1, 1 $	$3$	$4$	$( 4, 6, 5, 7)( 8,11, 9,10)$
$ 4, 4, 1, 1, 1, 1 $	$3$	$4$	$( 4, 7, 5, 6)( 8,10, 9,11)$
$ 8, 2, 1, 1 $	$12$	$8$	$( 2, 3)( 4, 8, 6,11, 5, 9, 7,10)$
$ 8, 2, 1, 1 $	$12$	$8$	$( 2, 3)( 4, 8, 7,10, 5, 9, 6,11)$
$ 8, 2, 1, 1 $	$12$	$8$	$( 2, 3)( 4,10, 7, 9, 5,11, 6, 8)$
$ 8, 2, 1, 1 $	$12$	$8$	$( 2, 3)( 4,10, 6, 8, 5,11, 7, 9)$
$ 2, 2, 2, 2, 2, 2 $	$12$	$2$	$( 1, 2)( 3,12)( 4, 8)( 5, 9)( 6,11)( 7,10)$
$ 4, 4, 2, 2 $	$12$	$4$	$( 1, 2)( 3,12)( 4, 8, 5, 9)( 6,11, 7,10)$
$ 2, 2, 2, 2, 2, 2 $	$12$	$2$	$( 1, 2)( 3,12)( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 4, 4, 2, 2 $	$12$	$4$	$( 1, 2)( 3,12)( 4,10, 5,11)( 6, 8, 7, 9)$
$ 4, 4, 2, 2 $	$3$	$4$	$( 1, 2,12, 3)( 4, 5)( 6, 7)( 8,10, 9,11)$
$ 4, 4, 2, 2 $	$6$	$4$	$( 1, 2,12, 3)( 4, 5)( 6, 7)( 8,11, 9,10)$
$ 4, 4, 2, 2 $	$3$	$4$	$( 1, 2,12, 3)( 4, 6, 5, 7)( 8, 9)(10,11)$
$ 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, 944]

Character table:

2 6 6 6 5 6 6 4 4 4 4 4 4 4 4 6 5 6 1 1 6 3 1 . . . . . . . . . . . . . . . . 1 1 1 1a 2a 2b 4a 4b 4c 8a 8b 8c 8d 2c 4d 2d 4e 4f 4g 4h 3a 6a 2e 2P 1a 1a 1a 2b 2b 2b 4b 4c 4c 4b 1a 2b 1a 2b 2b 2b 2b 3a 3a 1a 3P 1a 2a 2b 4a 4c 4b 8c 8d 8a 8b 2c 4d 2d 4e 4h 4g 4f 1a 2e 2e 5P 1a 2a 2b 4a 4b 4c 8a 8b 8c 8d 2c 4d 2d 4e 4f 4g 4h 3a 6a 2e 7P 1a 2a 2b 4a 4c 4b 8c 8d 8a 8b 2c 4d 2d 4e 4h 4g 4f 3a 6a 2e 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 3 3 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 . . 3 X.10 3 3 3 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 . . 3 X.11 3 -1 -1 1 A /A B -B -B B -1 1 -1 1 /A 1 A . . 3 X.12 3 -1 -1 1 /A A -B B B -B -1 1 -1 1 A 1 /A . . 3 X.13 3 -1 -1 1 A /A -B B B -B 1 -1 1 -1 /A 1 A . . 3 X.14 3 -1 -1 1 /A A B -B -B B 1 -1 1 -1 A 1 /A . . 3 X.15 3 1 -1 -1 A /A B B -B -B 1 1 -1 -1 -/A 1 -A . . -3 X.16 3 1 -1 -1 /A A -B -B B B 1 1 -1 -1 -A 1 -/A . . -3 X.17 3 1 -1 -1 A /A -B -B B B -1 -1 1 1 -/A 1 -A . . -3 X.18 3 1 -1 -1 /A A B B -B -B -1 -1 1 1 -A 1 -/A . . -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 A = -1-2*E(4) = -1-2*Sqrt(-1) = -1-2i B = -E(4) = -Sqrt(-1) = -i