Galois Group: 12T51

• Label: 12T51
• Order: \(96\)
• n: \(12\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No
• Group: $D_4\times A_4$

Group action invariants

Degree $n$ :		$12$
Transitive number $t$ :		$51$
Group :		$D_4\times A_4$
CHM label :		$[1/16.D(4)^{3}]3$
Parity:		$-1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(3,9)(6,12), (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (1,7)(3,9)(5,11), (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
3:	$C_3$
4:	$C_2^2$
6:	$C_6$ x 3
8:	$D_{4}$
12:	$A_4$, $C_6\times C_2$
24:	$A_4\times C_2$ x 3, $D_4 \times C_3$
48:	$C_2^2 \times A_4$

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

12T51, 16T179 x 2, 24T160, 24T161 x 2, 24T162 x 2, 24T163 x 2, 24T164 x 2, 32T385

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

Group invariants

Order:		$96=2^{5} \cdot 3$
Cyclic:		No
Abelian:		No
Solvable:		Yes
GAP id:		[96, 197]

Character table:

2 5 4 5 4 5 2 2 3 2 3 4 4 4 4 2 3 3 2 2 5 3 1 . . 1 . 1 1 1 1 1 . 1 . 1 1 1 1 1 1 1 1a 2a 2b 2c 2d 12a 6a 6b 6c 3a 2e 4a 4b 2f 6d 6e 3b 6f 12b 2g 2P 1a 1a 1a 1a 1a 6b 3a 3b 3b 3b 1a 2g 2g 1a 3a 3a 3a 3b 6e 1a 3P 1a 2a 2b 2c 2d 4a 2f 2g 2c 1a 2e 4a 4b 2f 2c 2g 1a 2f 4a 2g 5P 1a 2a 2b 2c 2d 12b 6f 6e 6d 3b 2e 4a 4b 2f 6c 6b 3a 6a 12a 2g 7P 1a 2a 2b 2c 2d 12a 6a 6b 6c 3a 2e 4a 4b 2f 6d 6e 3b 6f 12b 2g 11P 1a 2a 2b 2c 2d 12b 6f 6e 6d 3b 2e 4a 4b 2f 6c 6b 3a 6a 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 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 A -A -/A /A -/A 1 -1 -1 1 A -A -A -/A /A 1 X.6 1 -1 1 -1 1 /A -/A -A A -A 1 -1 -1 1 /A -/A -/A -A A 1 X.7 1 -1 1 -1 1 -/A /A -A A -A -1 1 1 -1 /A -/A -/A A -A 1 X.8 1 -1 1 -1 1 -A A -/A /A -/A -1 1 1 -1 A -A -A /A -/A 1 X.9 1 1 1 1 1 A A -/A -/A -/A -1 -1 -1 -1 -A -A -A /A /A 1 X.10 1 1 1 1 1 /A /A -A -A -A -1 -1 -1 -1 -/A -/A -/A A A 1 X.11 1 1 1 1 1 -/A -/A -A -A -A 1 1 1 1 -/A -/A -/A -A -A 1 X.12 1 1 1 1 1 -A -A -/A -/A -/A 1 1 1 1 -A -A -A -/A -/A 1 X.13 2 . -2 . 2 . . -2 . 2 . . . . . -2 2 . . -2 X.14 2 . -2 . 2 . . B . -B . . . . . /B -/B . . -2 X.15 2 . -2 . 2 . . /B . -/B . . . . . B -B . . -2 X.16 3 -1 -1 3 -1 . . . . . -1 3 -1 3 . . . . . 3 X.17 3 -1 -1 3 -1 . . . . . 1 -3 1 -3 . . . . . 3 X.18 3 1 -1 -3 -1 . . . . . -1 -3 1 3 . . . . . 3 X.19 3 1 -1 -3 -1 . . . . . 1 3 -1 -3 . . . . . 3 X.20 6 . 2 . -2 . . . . . . . . . . . . . . -6 A = -E(3) = (1-Sqrt(-3))/2 = -b3 B = -2*E(3)^2 = 1+Sqrt(-3) = 1+i3