Galois group: 12T41

• Label: 12T41
• Degree: $12$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_2\times C_3^2:C_4$

Group invariants

Abstract group:		$C_2\times C_3^2:C_4$
Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$41$
CHM label:		$1/2[(1/4.2^{3})^{2}]F_{36}(6)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,6,12,7)(2,8,11,5)(3,9,10,4)$, $(1,5,9)(4,8,12)$, $(1,12)(2,3)(4,5)(6,7)(8,9)(10,11)$, $(2,10)(3,11)(4,8)(5,9)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$8$:	$C_4\times C_2$
$36$:	$C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: $C_3^2:C_4$

Low degree siblings

12T40 x 2, 12T41, 18T27 x 2, 24T76 x 2, 36T35, 36T36

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	Index	Representative
1A	$1^{12}$	$1$	$1$	$0$	$()$
2A	$2^{6}$	$1$	$2$	$6$	$( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
2B	$2^{6}$	$9$	$2$	$6$	$( 1, 8)( 2, 3)( 4, 5)( 6,11)( 7,10)( 9,12)$
2C	$2^{4},1^{4}$	$9$	$2$	$4$	$( 1, 5)( 2, 6)( 3, 7)( 4,12)$
3A	$3^{2},1^{6}$	$4$	$3$	$4$	$( 1, 5, 9)( 4, 8,12)$
3B	$3^{4}$	$4$	$3$	$8$	$( 1, 9, 5)( 2, 6,10)( 3, 7,11)( 4,12, 8)$
4A1	$4^{3}$	$9$	$4$	$9$	$( 1, 7, 8,10)( 2, 5, 3, 4)( 6, 9,11,12)$
4A-1	$4^{3}$	$9$	$4$	$9$	$( 1,10, 8, 7)( 2, 4, 3, 5)( 6,12,11, 9)$
4B1	$4^{3}$	$9$	$4$	$9$	$( 1, 3, 8,10)( 2, 9,11,12)( 4, 6, 5, 7)$
4B-1	$4^{3}$	$9$	$4$	$9$	$( 1,10, 8, 3)( 2,12,11, 9)( 4, 7, 5, 6)$
6A	$6,2^{3}$	$4$	$6$	$8$	$( 1, 8, 5,12, 9, 4)( 2, 3)( 6, 7)(10,11)$
6B	$6^{2}$	$4$	$6$	$10$	$( 1, 4, 9,12, 5, 8)( 2,11, 6, 3,10, 7)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	3A	3B	4A1	4A-1	4B1	4B-1	6A	6B
	Size	1	1	9	9	4	4	9	9	9	9	4	4
	2 P	1A	1A	1A	1A	3A	3B	2B	2B	2B	2B	3A	3B
	3 P	1A	2A	2B	2C	1A	1A	4A-1	4A1	4B-1	4B1	2A	2A
	Type
72.45.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.45.1b	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$
72.45.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$
72.45.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
72.45.1e1	C	$1$	$-1$	$-1$	$1$	$1$	$1$	$-i$	$-i$	$i$	$i$	$-1$	$-1$
72.45.1e2	C	$1$	$-1$	$-1$	$1$	$1$	$1$	$i$	$i$	$-i$	$-i$	$-1$	$-1$
72.45.1f1	C	$1$	$1$	$-1$	$-1$	$1$	$1$	$-i$	$i$	$-i$	$i$	$1$	$1$
72.45.1f2	C	$1$	$1$	$-1$	$-1$	$1$	$1$	$i$	$-i$	$i$	$-i$	$1$	$1$
72.45.4a	R	$4$	$4$	$0$	$0$	$-2$	$1$	$0$	$0$	$0$	$0$	$-2$	$1$
72.45.4b	R	$4$	$4$	$0$	$0$	$1$	$-2$	$0$	$0$	$0$	$0$	$1$	$-2$
72.45.4c	R	$4$	$-4$	$0$	$0$	$-2$	$1$	$0$	$0$	$0$	$0$	$2$	$-1$
72.45.4d	R	$4$	$-4$	$0$	$0$	$1$	$-2$	$0$	$0$	$0$	$0$	$-1$	$2$

Regular extensions

Data not computed