Galois Group: 12T87

• Label: 12T87
• 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 $t$ :		$87$
Group :		$C_2\times C_2^2\wr C_2:C_3$
CHM label :		$[2^{5}]6$
Parity:		$1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,12)(2,3), (1,3,5,7,9,11)(2,4,6,8,10,12)
$|\Aut(F/K)|$:		$2$

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: $C_2$

Degree 3: $C_3$

Degree 4: None

Degree 6: $C_6$

Low degree siblings

12T87, 12T88 x 2, 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, 1, 1, 1, 1, 1, 1, 1, 1 $	$6$	$2$	$( 8, 9)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $	$6$	$2$	$( 6, 7)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $	$3$	$2$	$( 4, 5)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $	$6$	$2$	$( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $	$6$	$2$	$( 2, 3)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $	$3$	$2$	$( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 6, 6 $	$16$	$6$	$( 1, 2, 4, 6, 8,10)( 3, 5, 7, 9,11,12)$
$ 6, 6 $	$16$	$6$	$( 1, 2, 4, 6, 8,11)( 3, 5, 7, 9,10,12)$
$ 3, 3, 3, 3 $	$16$	$3$	$( 1, 4, 8)( 2, 6,10)( 3, 7,11)( 5, 9,12)$
$ 6, 6 $	$16$	$6$	$( 1, 4, 8,12, 5, 9)( 2, 6,10, 3, 7,11)$
$ 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1, 6)( 2, 8)( 3, 9)( 4,10)( 5,11)( 7,12)$
$ 4, 4, 2, 2 $	$12$	$4$	$( 1, 6)( 2, 8, 3, 9)( 4,10, 5,11)( 7,12)$
$ 4, 4, 2, 2 $	$12$	$4$	$( 1, 6,12, 7)( 2, 8)( 3, 9)( 4,10, 5,11)$
$ 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1, 6)( 2, 8)( 3, 9)( 4,11)( 5,10)( 7,12)$
$ 3, 3, 3, 3 $	$16$	$3$	$( 1, 8, 4)( 2,10, 6)( 3,11, 7)( 5,12, 9)$
$ 6, 6 $	$16$	$6$	$( 1, 8, 5,12, 9, 4)( 2,10, 7, 3,11, 6)$
$ 6, 6 $	$16$	$6$	$( 1,10, 8, 6, 4, 2)( 3,12,11, 9, 7, 5)$
$ 6, 6 $	$16$	$6$	$( 1,10, 9, 6, 4, 2)( 3,12,11, 8, 7, 5)$
$ 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 5 5 6 5 5 6 2 2 2 2 4 4 4 4 2 2 2 2 6 3 1 . . . . . . 1 1 1 1 1 . . 1 1 1 1 1 1 1a 2a 2b 2c 2d 2e 2f 6a 6b 3a 6c 2g 4a 4b 2h 3b 6d 6e 6f 2i 2P 1a 1a 1a 1a 1a 1a 1a 3a 3a 3b 3b 1a 2f 2f 1a 3a 3a 3b 3b 1a 3P 1a 2a 2b 2c 2d 2e 2f 2g 2h 1a 2i 2g 4a 4b 2h 1a 2i 2g 2h 2i 5P 1a 2a 2b 2c 2d 2e 2f 6e 6f 3b 6d 2g 4a 4b 2h 3a 6c 6a 6b 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 A -A -/A /A -1 1 -1 1 -A A /A -/A -1 X.6 1 -1 1 -1 1 -1 1 /A -/A -A A -1 1 -1 1 -/A /A A -A -1 X.7 1 -1 1 -1 1 -1 1 -/A /A -A A 1 -1 1 -1 -/A /A -A A -1 X.8 1 -1 1 -1 1 -1 1 -A A -/A /A 1 -1 1 -1 -A A -/A /A -1 X.9 1 1 1 1 1 1 1 A A -/A -/A -1 -1 -1 -1 -A -A /A /A 1 X.10 1 1 1 1 1 1 1 /A /A -A -A -1 -1 -1 -1 -/A -/A A A 1 X.11 1 1 1 1 1 1 1 -/A -/A -A -A 1 1 1 1 -/A -/A -A -A 1 X.12 1 1 1 1 1 1 1 -A -A -/A -/A 1 1 1 1 -A -A -/A -/A 1 X.13 3 -1 -1 3 -1 -1 3 . . . . -3 1 1 -3 . . . . 3 X.14 3 -1 -1 3 -1 -1 3 . . . . 3 -1 -1 3 . . . . 3 X.15 3 1 -1 -3 -1 1 3 . . . . -3 -1 1 3 . . . . -3 X.16 3 1 -1 -3 -1 1 3 . . . . 3 1 -1 -3 . . . . -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