Galois Group: 8T47

• Label: 8T47
• Order: \(1152\)
• n: \(8\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No
• Group: $S_4\wr C_2$

Group action invariants

Degree $n$ :		$8$
Transitive number $t$ :		$47$
Group :		$S_4\wr C_2$
CHM label :		$[S(4)^{2}]2$
Parity:		$-1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,2,3,8), (2,3), (1,5)(2,6)(3,7)(4,8)
$|\Aut(F/K)|$:		$1$

Low degree resolvents

|G/N|	Galois groups for stem field(s)
2:	$C_2$ x 3
4:	$C_2^2$
8:	$D_{4}$
72:	$C_3^2:D_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Low degree siblings

12T200, 12T201, 12T202, 12T203, 16T1292, 16T1294, 16T1295, 16T1296, 18T272, 18T273, 18T274, 18T275, 24T2803, 24T2804, 24T2805, 24T2806, 24T2807, 24T2808, 24T2809, 24T2810, 24T2821, 24T2826, 32T96692, 32T96694, 32T96695, 32T96696, 36T1758, 36T1759, 36T1760, 36T1761, 36T1762, 36T1763, 36T1764, 36T1765, 36T1766, 36T1767, 36T1768, 36T1769, 36T1943, 36T1944, 36T1945, 36T1946

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 $	$1$	$1$	$()$
$ 2, 2, 1, 1, 1, 1 $	$6$	$2$	$(1,3)(2,8)$
$ 2, 2, 2, 2 $	$9$	$2$	$(1,3)(2,8)(4,6)(5,7)$
$ 3, 1, 1, 1, 1, 1 $	$16$	$3$	$(2,3,8)$
$ 3, 2, 2, 1 $	$48$	$6$	$(2,3,8)(4,6)(5,7)$
$ 3, 3, 1, 1 $	$64$	$3$	$(2,3,8)(5,7,6)$
$ 2, 1, 1, 1, 1, 1, 1 $	$12$	$2$	$(3,8)$
$ 4, 1, 1, 1, 1 $	$12$	$4$	$(1,3,2,8)$
$ 2, 2, 2, 1, 1 $	$36$	$2$	$(3,8)(4,6)(5,7)$
$ 4, 2, 2 $	$36$	$4$	$(1,3,2,8)(4,6)(5,7)$
$ 3, 2, 1, 1, 1 $	$96$	$6$	$(3,8)(5,7,6)$
$ 4, 3, 1 $	$96$	$12$	$(1,3,2,8)(5,7,6)$
$ 2, 2, 1, 1, 1, 1 $	$36$	$2$	$(3,8)(6,7)$
$ 4, 2, 1, 1 $	$72$	$4$	$(1,3,2,8)(6,7)$
$ 4, 4 $	$36$	$4$	$(1,3,2,8)(4,6,5,7)$
$ 2, 2, 2, 2 $	$24$	$2$	$(1,5)(2,6)(3,7)(4,8)$
$ 4, 4 $	$72$	$4$	$(1,5,3,7)(2,6,8,4)$
$ 6, 2 $	$192$	$6$	$(1,5)(2,6,3,7,8,4)$
$ 4, 2, 2 $	$144$	$4$	$(1,5)(2,6)(3,7,8,4)$
$ 8 $	$144$	$8$	$(1,5,3,7,2,6,8,4)$

Group invariants

Order:		$1152=2^{7} \cdot 3^{2}$
Cyclic:		No
Abelian:		No
Solvable:		Yes
GAP id:		Data not available

Character table:

2 7 6 7 3 3 1 5 5 5 5 2 2 5 4 5 4 4 1 3 3 3 2 1 . 2 1 2 1 1 . . 1 1 . . . 1 . 1 . . 1a 2a 2b 3a 6a 3b 2c 4a 2d 4b 6b 12a 2e 4c 4d 2f 4e 6c 4f 8a 2P 1a 1a 1a 3a 3a 3b 1a 2a 1a 2a 3a 6a 1a 2a 2b 1a 2b 3b 2e 4d 3P 1a 2a 2b 1a 2a 1a 2c 4a 2d 4b 2c 4a 2e 4c 4d 2f 4e 2f 4f 8a 5P 1a 2a 2b 3a 6a 3b 2c 4a 2d 4b 6b 12a 2e 4c 4d 2f 4e 6c 4f 8a 7P 1a 2a 2b 3a 6a 3b 2c 4a 2d 4b 6b 12a 2e 4c 4d 2f 4e 6c 4f 8a 11P 1a 2a 2b 3a 6a 3b 2c 4a 2d 4b 6b 12a 2e 4c 4d 2f 4e 6c 4f 8a 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 . . . . . X.6 4 4 4 1 1 -2 2 2 2 2 -1 -1 . . . . . . . . X.7 4 4 4 1 1 -2 -2 -2 -2 -2 1 1 . . . . . . . . X.8 4 4 4 -2 -2 1 . . . . . . . . . -2 -2 1 . . X.9 4 4 4 -2 -2 1 . . . . . . . . . 2 2 -1 . . X.10 6 2 -2 3 -1 . -4 -2 . 2 -1 1 2 . -2 . . . . . X.11 6 2 -2 3 -1 . -2 -4 2 . 1 -1 -2 . 2 . . . . . X.12 6 2 -2 3 -1 . 4 2 . -2 1 -1 2 . -2 . . . . . X.13 6 2 -2 3 -1 . 2 4 -2 . -1 1 -2 . 2 . . . . . X.14 9 -3 1 . . . -3 3 1 -1 . . 1 -1 1 -3 1 . 1 -1 X.15 9 -3 1 . . . -3 3 1 -1 . . 1 -1 1 3 -1 . -1 1 X.16 9 -3 1 . . . 3 -3 -1 1 . . 1 -1 1 -3 1 . -1 1 X.17 9 -3 1 . . . 3 -3 -1 1 . . 1 -1 1 3 -1 . 1 -1 X.18 12 4 -4 -3 1 . -2 2 -2 2 1 -1 . . . . . . . . X.19 12 4 -4 -3 1 . 2 -2 2 -2 -1 1 . . . . . . . . X.20 18 -6 2 . . . . . . . . . -2 2 -2 . . . . .