Galois group: 16T375

• Label: 16T375
• Degree: $16$
• Order: $128$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: yes
• Group: $C_4^2.(C_2\times C_4)$

Group invariants

Abstract group:		$C_4^2.(C_2\times C_4)$
Order:		$128=2^{7}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$5$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$375$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,14,2,13)(3,12,7,16)(4,11,8,15)(5,10)(6,9)$, $(1,4,2,3)(5,7,6,8)(9,16)(10,15)(11,13)(12,14)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$8$:	$D_{4}$ x 2, $C_4\times C_2$
$16$:	$C_2^2:C_4$
$32$:	$C_2^3 : C_4$
$64$:	16T154

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $C_2^3: C_4$

Low degree siblings

16T375, 32T850 x 2, 32T851, 32T1824 x 2, 32T2008

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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	$2^{4},1^{8}$	$2$	$2$	$4$	$(1,2)(3,4)(5,6)(7,8)$
2C	$2^{6},1^{4}$	$8$	$2$	$6$	$( 1, 6)( 2, 5)( 7, 8)(11,16)(12,15)(13,14)$
2D	$2^{6},1^{4}$	$8$	$2$	$6$	$( 1, 6)( 2, 5)( 3, 4)( 9,14)(10,13)(11,12)$
4A	$4^{4}$	$4$	$4$	$12$	$( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,13,10,14)(11,16,12,15)$
4B	$4^{2},2^{2},1^{4}$	$8$	$4$	$8$	$( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,10)(13,14)$
4C	$4^{2},2^{4}$	$16$	$4$	$10$	$( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12,10,11)(13,16,14,15)$
4D1	$4^{3},2^{2}$	$16$	$4$	$11$	$( 1,16, 6,11)( 2,15, 5,12)( 3,10)( 4, 9)( 7,14, 8,13)$
4D-1	$4^{3},2^{2}$	$16$	$4$	$11$	$( 1,11, 6,16)( 2,12, 5,15)( 3,10)( 4, 9)( 7,13, 8,14)$
4E1	$4^{3},2^{2}$	$16$	$4$	$11$	$( 1,13, 6,10)( 2,14, 5, 9)( 3,12, 4,11)( 7,16)( 8,15)$
4E-1	$4^{3},2^{2}$	$16$	$4$	$11$	$( 1,10, 6,13)( 2, 9, 5,14)( 3,11, 4,12)( 7,16)( 8,15)$
8A1	$8^{2}$	$8$	$8$	$14$	$( 1, 8, 5, 3, 2, 7, 6, 4)( 9,12,13,15,10,11,14,16)$
8A-1	$8^{2}$	$8$	$8$	$14$	$( 1, 3, 6, 8, 2, 4, 5, 7)( 9,15,14,12,10,16,13,11)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	2D	4A	4B	4C	4D1	4D-1	4E1	4E-1	8A1	8A-1
	Size	1	1	2	8	8	4	8	16	16	16	16	16	8	8
	2 P	1A	1A	1A	1A	1A	2A	2B	2B	2C	2C	2D	2D	4A	4A
	Type
128.144.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
128.144.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$
128.144.1c	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$
128.144.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$-1$
128.144.1e1	C	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-i$	$-1$	$i$	$i$	$-i$	$1$	$1$
128.144.1e2	C	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$i$	$-1$	$-i$	$-i$	$i$	$1$	$1$
128.144.1f1	C	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-i$	$1$	$-i$	$i$	$i$	$-1$	$-1$
128.144.1f2	C	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$i$	$1$	$i$	$-i$	$-i$	$-1$	$-1$
128.144.2a	R	$2$	$2$	$2$	$-2$	$2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.144.2b	R	$2$	$2$	$2$	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.144.4a	R	$4$	$4$	$4$	$0$	$0$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.144.4b1	C	$4$	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-2i$	$2i$
128.144.4b2	C	$4$	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$2i$	$-2i$
128.144.8a	R	$8$	$-8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed