Galois Group: 12T105

• Label: 12T105
• Order: \(192\)
• n: \(12\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No
• Group: $C_2^5.C_6$

Group action invariants

Degree $n$ :		$12$
Transitive number $t$ :		$105$
Group :		$C_2^5.C_6$
CHM label :		$1/2[2^{6}]6$
Parity:		$-1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,12)(2,3), (1,2,4,6,8,10,12,3,5,7,9,11)
$|\Aut(F/K)|$:		$2$

Low degree resolvents

|G/N|	Galois groups for stem field(s)
2:	$C_2$
3:	$C_3$
4:	$C_4$
6:	$C_6$
12:	$A_4$, $C_{12}$
24:	$A_4\times C_2$
48:	12T29
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

12T99 x 2, 12T105, 16T418 x 2, 24T483 x 2, 24T484 x 2, 24T501 x 2, 24T502 x 4, 24T503 x 2, 24T504 x 2, 24T505 x 2, 24T506 x 2, 24T507 x 4

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 $	$3$	$2$	$( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 12 $	$16$	$12$	$( 1, 2, 4, 6, 8,10,12, 3, 5, 7, 9,11)$
$ 12 $	$16$	$12$	$( 1, 2, 4, 6, 8,11,12, 3, 5, 7, 9,10)$
$ 3, 3, 3, 3 $	$16$	$3$	$( 1, 4, 8)( 2, 6,10)( 3, 7,11)( 5, 9,12)$
$ 6, 6 $	$16$	$6$	$( 1, 4, 8,12, 5, 9)( 2, 6,10, 3, 7,11)$
$ 4, 2, 2, 2, 2 $	$12$	$4$	$( 1, 6)( 2, 8)( 3, 9)( 4,10, 5,11)( 7,12)$
$ 4, 2, 2, 2, 2 $	$12$	$4$	$( 1, 6)( 2, 8, 3, 9)( 4,10)( 5,11)( 7,12)$
$ 4, 4, 4 $	$4$	$4$	$( 1, 6,12, 7)( 2, 8, 3, 9)( 4,10, 5,11)$
$ 4, 4, 4 $	$4$	$4$	$( 1, 6,12, 7)( 2, 8, 3, 9)( 4,11, 5,10)$
$ 3, 3, 3, 3 $	$16$	$3$	$( 1, 8, 4)( 2,10, 6)( 3,11, 7)( 5,12, 9)$
$ 6, 6 $	$16$	$6$	$( 1, 8, 5,12, 9, 4)( 2,10, 7, 3,11, 6)$
$ 12 $	$16$	$12$	$( 1,10, 9, 7, 5, 3,12,11, 8, 6, 4, 2)$
$ 12 $	$16$	$12$	$( 1,10, 8, 7, 5, 3,12,11, 9, 6, 4, 2)$
$ 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, 191]

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 12a 12b 3a 6a 4a 4b 4c 4d 3b 6b 12c 12d 2g 2P 1a 1a 1a 1a 1a 1a 1a 6a 6a 3b 3b 2c 2c 2g 2g 3a 3a 6b 6b 1a 3P 1a 2a 2b 2c 2d 2e 2f 4c 4d 1a 2g 4b 4a 4d 4c 1a 2g 4c 4d 2g 5P 1a 2a 2b 2c 2d 2e 2f 12c 12d 3b 6b 4a 4b 4c 4d 3a 6a 12a 12b 2g 7P 1a 2a 2b 2c 2d 2e 2f 12b 12a 3a 6a 4b 4a 4d 4c 3b 6b 12d 12c 2g 11P 1a 2a 2b 2c 2d 2e 2f 12d 12c 3b 6b 4b 4a 4d 4c 3a 6a 12b 12a 2g 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 A -A 1 -1 A -A -A A 1 -1 A -A -1 X.4 1 -1 1 -1 1 -1 1 -A A 1 -1 -A A A -A 1 -1 -A A -1 X.5 1 -1 1 -1 1 -1 1 B -B -/C /C A -A -A A -C C -/B /B -1 X.6 1 -1 1 -1 1 -1 1 -/B /B -C C A -A -A A -/C /C B -B -1 X.7 1 -1 1 -1 1 -1 1 /B -/B -C C -A A A -A -/C /C -B B -1 X.8 1 -1 1 -1 1 -1 1 -B B -/C /C -A A A -A -C C /B -/B -1 X.9 1 1 1 1 1 1 1 C C -/C -/C -1 -1 -1 -1 -C -C /C /C 1 X.10 1 1 1 1 1 1 1 /C /C -C -C -1 -1 -1 -1 -/C -/C C C 1 X.11 1 1 1 1 1 1 1 -/C -/C -C -C 1 1 1 1 -/C -/C -C -C 1 X.12 1 1 1 1 1 1 1 -C -C -/C -/C 1 1 1 1 -C -C -/C -/C 1 X.13 3 -1 -1 3 -1 -1 3 . . . . -1 -1 3 3 . . . . 3 X.14 3 -1 -1 3 -1 -1 3 . . . . 1 1 -3 -3 . . . . 3 X.15 3 1 -1 -3 -1 1 3 . . . . A -A D -D . . . . -3 X.16 3 1 -1 -3 -1 1 3 . . . . -A A -D D . . . . -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(4) = -Sqrt(-1) = -i B = -E(12)^7 C = -E(3) = (1-Sqrt(-3))/2 = -b3 D = -3*E(4) = -3*Sqrt(-1) = -3i