Galois group: 12T47

• Label: 12T47
• Degree: $12$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $\PSU(3,2)$

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$47$
Group:		$\PSU(3,2)$
CHM label:		$[(1/3.3^{3}):2]E(4)_{4}$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,5,10)(3,12)(4,11,8,7)(6,9), (1,5)(2,10)(4,8)(7,11), (1,4,5,8)(2,7,10,11)(3,6)(9,12), (2,6,10)(3,7,11)(4,8,12)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$Q_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: None

Low degree siblings

9T14, 18T35 x 3, 24T82, 36T55

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Label	Cycle Type	Size	Order	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 1, 1, 1, 1 $	$9$	$2$	$( 3,11)( 4, 8)( 5, 9)( 6,10)$
	$ 3, 3, 3, 1, 1, 1 $	$8$	$3$	$( 2, 6,10)( 3, 7,11)( 4, 8,12)$
	$ 4, 4, 2, 2 $	$18$	$4$	$( 1, 2)( 3, 4,11, 8)( 5, 6, 9,10)( 7,12)$
	$ 4, 4, 2, 2 $	$18$	$4$	$( 1, 3)( 2, 4, 6,12)( 5,11, 9, 7)( 8,10)$
	$ 4, 4, 2, 2 $	$18$	$4$	$( 1, 4, 9, 8)( 2, 3, 6, 7)( 5,12)(10,11)$

Group invariants

Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		72.41
Character table:

		1A	2A	3A	4A	4B	4C
	Size	1	9	8	18	18	18
	2 P	1A	1A	3A	2A	2A	2A
	3 P	1A	2A	1A	4A	4B	4C
	Type
72.41.1a	R	$1$	$1$	$1$	$1$	$1$	$1$
72.41.1b	R	$1$	$1$	$1$	$-1$	$-1$	$1$
72.41.1c	R	$1$	$1$	$1$	$-1$	$1$	$-1$
72.41.1d	R	$1$	$1$	$1$	$1$	$-1$	$-1$
72.41.2a	S	$2$	$-2$	$2$	$0$	$0$	$0$
72.41.8a	R	$8$	$0$	$-1$	$0$	$0$	$0$

Additional information

This is an elusive group.