Galois group: 16T193

• Label: 16T193
• Degree: $16$
• Order: $96$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $\GL(2,\mathbb{Z}/4)$

Group invariants

Abstract group:		$\GL(2,\mathbb{Z}/4)$
Order:		$96=2^{5} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$193$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,14,8,15)(2,13,7,16)(3,9,5,12)(4,10,6,11)$, $(1,8,4)(2,7,3)(9,14,12,10,13,11)(15,16)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$6$:	$S_3$
$8$:	$D_{4}$
$12$:	$D_{6}$
$24$:	$S_4$, $(C_6\times C_2):C_2$
$48$:	$S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$, $S_4$

Degree 8: $S_4$

Low degree siblings

12T49 x 2, 12T50, 12T52, 16T186, 24T153, 24T154, 24T155 x 2, 24T156, 24T157, 24T158, 24T159, 24T165, 24T166, 32T392

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	$1^{16}$	$1$	$1$	$0$	$()$
2A	$2^{8}$	$1$	$2$	$8$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2B	$2^{4},1^{8}$	$2$	$2$	$4$	$( 9,10)(11,12)(13,14)(15,16)$
2C	$2^{8}$	$3$	$2$	$8$	$( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,16)(10,15)(11,14)(12,13)$
2D	$2^{8}$	$3$	$2$	$8$	$( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,14)(10,13)(11,16)(12,15)$
2E	$2^{8}$	$6$	$2$	$8$	$( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,11)(10,12)(13,15)(14,16)$
2F	$2^{8}$	$12$	$2$	$8$	$( 1,12)( 2,11)( 3,14)( 4,13)( 5,10)( 6, 9)( 7,15)( 8,16)$
3A	$3^{4},1^{4}$	$8$	$3$	$8$	$( 1, 8, 4)( 2, 7, 3)( 9,13,12)(10,14,11)$
4A	$4^{4}$	$12$	$4$	$12$	$( 1,11, 2,12)( 3,16, 4,15)( 5,13, 6,14)( 7, 9, 8,10)$
4B	$4^{4}$	$12$	$4$	$12$	$( 1,10, 4,15)( 2, 9, 3,16)( 5,12, 7,13)( 6,11, 8,14)$
4C	$4^{4}$	$12$	$4$	$12$	$( 1,10, 5,13)( 2, 9, 6,14)( 3,12, 8,15)( 4,11, 7,16)$
6A	$6^{2},2^{2}$	$8$	$6$	$12$	$( 1, 3, 8, 2, 4, 7)( 5, 6)( 9,11,13,10,12,14)(15,16)$
6B1	$6,3^{2},2,1^{2}$	$8$	$6$	$10$	$( 1, 8, 4)( 2, 7, 3)( 9,14,12,10,13,11)(15,16)$
6B-1	$6,3^{2},2,1^{2}$	$8$	$6$	$10$	$( 1, 7, 4, 2, 8, 3)( 5, 6)( 9,13,12)(10,14,11)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	2D	2E	2F	3A	4A	4B	4C	6A	6B1	6B-1
	Size	1	1	2	3	3	6	12	8	12	12	12	8	8	8
	2 P	1A	1A	1A	1A	1A	1A	1A	3A	2A	2C	2D	3A	3A	3A
	3 P	1A	2A	2B	2C	2D	2E	2F	1A	4A	4B	4C	2A	2B	2B
	Type
96.195.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
96.195.1b	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$
96.195.1c	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$
96.195.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$
96.195.2a	R	$2$	$2$	$2$	$2$	$2$	$2$	$0$	$-1$	$0$	$0$	$0$	$-1$	$-1$	$-1$
96.195.2b	R	$2$	$-2$	$0$	$2$	$-2$	$0$	$0$	$2$	$0$	$0$	$0$	$0$	$0$	$-2$
96.195.2c	R	$2$	$2$	$-2$	$2$	$2$	$-2$	$0$	$-1$	$0$	$0$	$0$	$1$	$1$	$-1$
96.195.2d1	C	$2$	$-2$	$0$	$2$	$-2$	$0$	$0$	$-1$	$0$	$0$	$0$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$1$
96.195.2d2	C	$2$	$-2$	$0$	$2$	$-2$	$0$	$0$	$-1$	$0$	$0$	$0$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$1$
96.195.3a	R	$3$	$3$	$3$	$-1$	$-1$	$-1$	$1$	$0$	$1$	$-1$	$-1$	$0$	$0$	$0$
96.195.3b	R	$3$	$3$	$3$	$-1$	$-1$	$-1$	$-1$	$0$	$-1$	$1$	$1$	$0$	$0$	$0$
96.195.3c	R	$3$	$3$	$-3$	$-1$	$-1$	$1$	$-1$	$0$	$1$	$1$	$-1$	$0$	$0$	$0$
96.195.3d	R	$3$	$3$	$-3$	$-1$	$-1$	$1$	$1$	$0$	$-1$	$-1$	$1$	$0$	$0$	$0$
96.195.6a	R	$6$	$-6$	$0$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed