LMFDB - Galois group: 32T2064

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magma:G := TransitiveGroup(32, 2064);

Group invariants

Abstract group:	$C_4^2.(C_2\times C_4)$	magma:IdentifyGroup(G);
Order:	$128=2^{7}$	magma:Order(G);
Cyclic:	no	magma:IsCyclic(G);
Abelian:	no	magma:IsAbelian(G);
Solvable:	yes	magma:IsSolvable(G);
Nilpotency class:	$4$	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$32$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$2064$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$8$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,23,20,6,2,24,19,5)(3,22,18,8,4,21,17,7)(9,32,26,13,10,31,25,14)(11,29,28,15,12,30,27,16)$, $(1,27,2,28)(3,25,4,26)(5,14,6,13)(7,15,8,16)(9,18,10,17)(11,20,12,19)(21,29,22,30)(23,32,24,31)$, $(1,22,20,7,2,21,19,8)(3,24,18,6,4,23,17,5)(9,30,26,15,10,29,25,16)(11,32,28,14,12,31,27,13)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 7
$4$: $C_4$ x 4, $C_2^2$ x 7
$8$: $D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$16$: $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
$32$: $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$
$64$: 16T76

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_4$ x 4, $C_2^2$ x 7
$8$:	$D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$16$:	$D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
$32$:	$C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$
$64$:	16T76

Subfields

Degree 2: $C_2$ x 3
Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 4
Degree 8: $C_4\times C_2$, $D_4$ x 2, $C_2^2:C_4$ x 2
Degree 16: $C_2^2 : C_4$

Low degree siblings

32T1854, 32T1861, 32T1865 x 2
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	$1^{32}$	$1$	$1$	$0$	$()$
2A	$2^{16}$	$1$	$2$	$16$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2B	$2^{8},1^{16}$	$2$	$2$	$8$	$( 9,10)(11,12)(13,14)(15,16)(25,26)(27,28)(29,30)(31,32)$
2C	$2^{8},1^{16}$	$2$	$2$	$8$	$( 5, 6)( 7, 8)( 9,10)(11,12)(21,22)(23,24)(25,26)(27,28)$
2D	$2^{8},1^{16}$	$2$	$2$	$8$	$( 5, 6)( 7, 8)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)$
2E	$2^{16}$	$8$	$2$	$16$	$( 1,28)( 2,27)( 3,26)( 4,25)( 5,14)( 6,13)( 7,15)( 8,16)( 9,18)(10,17)(11,20)(12,19)(21,29)(22,30)(23,32)(24,31)$
4A	$4^{8}$	$4$	$4$	$24$	$( 1,17, 2,18)( 3,20, 4,19)( 5,21, 6,22)( 7,23, 8,24)( 9,27,10,28)(11,26,12,25)(13,29,14,30)(15,31,16,32)$
4B	$4^{8}$	$4$	$4$	$24$	$( 1,17, 2,18)( 3,20, 4,19)( 5,21, 6,22)( 7,23, 8,24)( 9,28,10,27)(11,25,12,26)(13,30,14,29)(15,32,16,31)$
4C	$4^{8}$	$4$	$4$	$24$	$( 1,20, 2,19)( 3,18, 4,17)( 5,23, 6,24)( 7,22, 8,21)( 9,25,10,26)(11,27,12,28)(13,32,14,31)(15,29,16,30)$
4D	$4^{8}$	$4$	$4$	$24$	$( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,20,18,19)(21,23,22,24)(25,28,26,27)(29,32,30,31)$
4E	$4^{8}$	$4$	$4$	$24$	$( 1,20, 2,19)( 3,18, 4,17)( 5,23, 6,24)( 7,22, 8,21)( 9,26,10,25)(11,28,12,27)(13,31,14,32)(15,30,16,29)$
4F	$4^{8}$	$4$	$4$	$24$	$( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,24,22,23)(25,27,26,28)(29,32,30,31)$
4G	$4^{4},2^{8}$	$8$	$4$	$20$	$( 1,12)( 2,11)( 3,10)( 4, 9)( 5,31, 6,32)( 7,29, 8,30)(13,23,14,24)(15,21,16,22)(17,25)(18,26)(19,27)(20,28)$
4H	$4^{4},2^{8}$	$8$	$4$	$20$	$( 1,12, 2,11)( 3,10, 4, 9)( 5,31)( 6,32)( 7,29)( 8,30)(13,24)(14,23)(15,22)(16,21)(17,25,18,26)(19,27,20,28)$
4I	$4^{8}$	$8$	$4$	$24$	$( 1,28, 2,27)( 3,26, 4,25)( 5,14, 6,13)( 7,15, 8,16)( 9,17,10,18)(11,19,12,20)(21,29,22,30)(23,32,24,31)$
8A1	$8^{4}$	$8$	$8$	$28$	$( 1,23,20, 6, 2,24,19, 5)( 3,22,18, 8, 4,21,17, 7)( 9,32,26,13,10,31,25,14)(11,29,28,15,12,30,27,16)$
8A-1	$8^{4}$	$8$	$8$	$28$	$( 1, 6,19,23, 2, 5,20,24)( 3, 8,17,22, 4, 7,18,21)( 9,13,25,32,10,14,26,31)(11,15,27,29,12,16,28,30)$
8B1	$8^{4}$	$8$	$8$	$28$	$( 1,14, 4,16, 2,13, 3,15)( 5,12, 7,10, 6,11, 8, 9)(17,29,20,32,18,30,19,31)(21,25,24,27,22,26,23,28)$
8B-1	$8^{4}$	$8$	$8$	$28$	$( 1,32, 3,30, 2,31, 4,29)( 5,28, 7,26, 6,27, 8,25)( 9,23,11,21,10,24,12,22)(13,17,15,19,14,18,16,20)$
8C1	$8^{4}$	$8$	$8$	$28$	$( 1,23,20, 6, 2,24,19, 5)( 3,22,18, 8, 4,21,17, 7)( 9,31,26,14,10,32,25,13)(11,30,28,16,12,29,27,15)$
8C-1	$8^{4}$	$8$	$8$	$28$	$( 1, 6,19,23, 2, 5,20,24)( 3, 8,17,22, 4, 7,18,21)( 9,14,25,31,10,13,26,32)(11,16,27,30,12,15,28,29)$
8D1	$8^{4}$	$8$	$8$	$28$	$( 1,14, 3,15, 2,13, 4,16)( 5,12, 8, 9, 6,11, 7,10)(17,29,19,31,18,30,20,32)(21,25,23,28,22,26,24,27)$
8D-1	$8^{4}$	$8$	$8$	$28$	$( 1,32, 4,29, 2,31, 3,30)( 5,28, 8,25, 6,27, 7,26)( 9,24,12,21,10,23,11,22)(13,18,16,19,14,17,15,20)$

Malle's constant $a(G)$: $1/8$

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A1	8A-1	8B1	8B-1	8C1	8C-1	8D1	8D-1
	Size	1	1	2	2	2	8	4	4	4	4	4	4	8	8	8	8	8	8	8	8	8	8	8
	2 P	1A	1A	1A	1A	1A	1A	2A	2A	2A	2A	2A	2A	2D	2D	2A	4E	4E	4F	4F	4E	4E	4F	4F
	Type
128.865.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
128.865.1b	R	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$
128.865.1c	R	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$
128.865.1d	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$
128.865.1e	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
128.865.1f	R	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$
128.865.1g	R	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$
128.865.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
128.865.1i1	C	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$- i$	$- i$	$i$	$i$	$i$	$- i$	$- i$	$i$
128.865.1i2	C	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$i$	$i$	$- i$	$- i$	$- i$	$i$	$i$	$- i$
128.865.1j1	C	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- i$	$i$	$- i$	$i$	$i$	$i$	$- i$	$- i$
128.865.1j2	C	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$i$	$- i$	$i$	$- i$	$- i$	$- i$	$i$	$i$
128.865.1k1	C	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- i$	$- i$	$i$	$i$	$- i$	$i$	$i$	$- i$
128.865.1k2	C	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$i$	$i$	$- i$	$- i$	$i$	$- i$	$- i$	$i$
128.865.1l1	C	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$- i$	$i$	$- i$	$i$	$- i$	$- i$	$i$	$i$
128.865.1l2	C	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$i$	$- i$	$i$	$- i$	$i$	$i$	$- i$	$- i$
128.865.2a	R	$2$	$2$	$- 2$	$- 2$	$2$	$0$	$- 2$	$- 2$	$2$	$2$	$2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.865.2b	R	$2$	$2$	$- 2$	$- 2$	$2$	$0$	$2$	$- 2$	$- 2$	$2$	$- 2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.865.2c	R	$2$	$2$	$2$	$2$	$2$	$0$	$- 2$	$- 2$	$- 2$	$- 2$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.865.2d	R	$2$	$2$	$2$	$2$	$2$	$0$	$2$	$- 2$	$2$	$- 2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.865.4a	R	$4$	$4$	$- 4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.865.4b	R	$4$	$4$	$4$	$- 4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.865.8a	S	$8$	$- 8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

magma:CharacterTable(G);

Regular extensions

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	32T2064

Degree	$32$
Order	$128$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$C_4^2.(C_2\times C_4)$