Galois Group: 12T88

• Label: 12T88
• Order: \(192\)
• n: \(12\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No
• Group: $C_2\times C_2^2\wr C_2:C_3$

Group action invariants

Degree $n$ :		$12$
Transitive number $t$ :		$88$
Group :		$C_2\times C_2^2\wr C_2:C_3$
CHM label :		$[2^{4}]A_{4}(6)$
Parity:		$-1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,12)(2,3)(4,5), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,12)(6,7), (1,3,11)(2,10,12)(4,6,8)(5,7,9)
$|\Aut(F/K)|$:		$2$

Low degree resolvents

|G/N|	Galois groups for stem field(s)
2:	$C_2$ x 3
3:	$C_3$
4:	$C_2^2$
6:	$C_6$ x 3
12:	$A_4$, $C_6\times C_2$
24:	$A_4\times C_2$ x 3
48:	$C_2^2 \times A_4$
96:	$C_2^4:C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$

Low degree siblings

12T87 x 2, 12T88, 16T416 x 2, 24T441 x 2, 24T442 x 2, 24T443 x 4, 24T444 x 2, 24T445 x 2, 24T446 x 2, 24T447 x 2, 24T448 x 4, 24T449, 24T450, 24T451 x 4, 24T452 x 4, 32T2183 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, 2, 1, 1, 1, 1, 1, 1 $	$4$	$2$	$( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $	$3$	$2$	$( 4, 5)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$4$	$2$	$( 4, 5)( 6, 7)( 8, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $	$3$	$2$	$( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $	$6$	$2$	$( 2, 8)( 3, 9)( 4,10)( 5,11)$
$ 4, 4, 2, 1, 1 $	$12$	$4$	$( 2, 8, 3, 9)( 4,10, 5,11)( 6, 7)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $	$6$	$2$	$( 2, 8)( 3, 9)( 4,11)( 5,10)$
$ 4, 4, 2, 1, 1 $	$12$	$4$	$( 2, 8, 3, 9)( 4,11, 5,10)( 6, 7)$
$ 6, 3, 3 $	$16$	$6$	$( 1, 2, 4)( 3, 5,12)( 6, 8,11, 7, 9,10)$
$ 3, 3, 3, 3 $	$16$	$3$	$( 1, 2, 4)( 3, 5,12)( 6, 9,11)( 7, 8,10)$
$ 6, 3, 3 $	$16$	$6$	$( 1, 2, 4,12, 3, 5)( 6, 8,11)( 7, 9,10)$
$ 6, 6 $	$16$	$6$	$( 1, 2, 4,12, 3, 5)( 6, 9,11, 7, 8,10)$
$ 6, 3, 3 $	$16$	$6$	$( 1, 4, 2)( 3,12, 5)( 6,10, 9, 7,11, 8)$
$ 3, 3, 3, 3 $	$16$	$3$	$( 1, 4, 2)( 3,12, 5)( 6,11, 9)( 7,10, 8)$
$ 6, 3, 3 $	$16$	$6$	$( 1, 4, 3,12, 5, 2)( 6,10, 8)( 7,11, 9)$
$ 6, 6 $	$16$	$6$	$( 1, 4, 3,12, 5, 2)( 6,11, 8, 7,10, 9)$
$ 2, 2, 2, 2, 2, 2 $	$6$	$2$	$( 1, 6)( 2, 3)( 4,10)( 5,11)( 7,12)( 8, 9)$
$ 2, 2, 2, 2, 2, 2 $	$6$	$2$	$( 1, 6)( 2, 3)( 4,11)( 5,10)( 7,12)( 8, 9)$
$ 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, 1000]

Character table:

2 6 4 6 4 6 5 4 5 4 2 2 2 2 2 2 2 2 5 5 6 3 1 1 . 1 . . . . . 1 1 1 1 1 1 1 1 . . 1 1a 2a 2b 2c 2d 2e 4a 2f 4b 6a 3a 6b 6c 6d 3b 6e 6f 2g 2h 2i 2P 1a 1a 1a 1a 1a 1a 2d 1a 2d 3b 3b 3b 3b 3a 3a 3a 3a 1a 1a 1a 3P 1a 2a 2b 2c 2d 2e 4a 2f 4b 2a 1a 2c 2i 2a 1a 2c 2i 2g 2h 2i 5P 1a 2a 2b 2c 2d 2e 4a 2f 4b 6d 3b 6e 6f 6a 3a 6b 6c 2g 2h 2i 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 1 -1 -1 1 A -A -A A /A -/A -/A /A 1 -1 -1 X.6 1 -1 -1 1 1 1 -1 -1 1 /A -/A -/A /A A -A -A A 1 -1 -1 X.7 1 -1 1 -1 1 1 -1 1 -1 A -A A -A /A -/A /A -/A 1 1 1 X.8 1 -1 1 -1 1 1 -1 1 -1 /A -/A /A -/A A -A A -A 1 1 1 X.9 1 1 -1 -1 1 1 1 -1 -1 -/A -/A /A /A -A -A A A 1 -1 -1 X.10 1 1 -1 -1 1 1 1 -1 -1 -A -A A A -/A -/A /A /A 1 -1 -1 X.11 1 1 1 1 1 1 1 1 1 -/A -/A -/A -/A -A -A -A -A 1 1 1 X.12 1 1 1 1 1 1 1 1 1 -A -A -A -A -/A -/A -/A -/A 1 1 1 X.13 3 -3 -3 3 3 -1 1 1 -1 . . . . . . . . -1 1 -3 X.14 3 -3 3 -3 3 -1 1 -1 1 . . . . . . . . -1 -1 3 X.15 3 3 -3 -3 3 -1 -1 1 1 . . . . . . . . -1 1 -3 X.16 3 3 3 3 3 -1 -1 -1 -1 . . . . . . . . -1 -1 3 X.17 6 . -2 . -2 -2 . 2 . . . . . . . . . 2 -2 6 X.18 6 . -2 . -2 2 . -2 . . . . . . . . . -2 2 6 X.19 6 . 2 . -2 -2 . -2 . . . . . . . . . 2 2 -6 X.20 6 . 2 . -2 2 . 2 . . . . . . . . . -2 -2 -6 A = -E(3) = (1-Sqrt(-3))/2 = -b3