LMFDB - Galois group: 32T3266

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magma: G := TransitiveGroup(32, 3266);

Group action invariants

Degree $n$:	$32$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$3266$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$D_4^2.C_4$
Parity:	$1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$4$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,13,3,15)(2,14,4,16)(5,11,7,10)(6,12,8,9)(17,23,26,32,19,21,27,29)(18,24,25,31,20,22,28,30), (1,31,15,27,12,24,8,19,3,30,13,26,9,22,6,17)(2,32,16,28,11,23,7,20,4,29,14,25,10,21,5,18)	magma: Generators(G);

Low degree resolvents

|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 $
$64$: $((C_8 : C_2):C_2):C_2$
$128$: 16T335

Resolvents shown for degrees $\leq 47$

\|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 $
$64$:	$((C_8 : C_2):C_2):C_2$
$128$:	16T335

Subfields

Degree 2: $C_2$
Degree 4: $C_4$, $D_{4}$ x 2
Degree 8: $C_2^2:C_4$, $((C_8 : C_2):C_2):C_2$ x 2
Degree 16: 16T130, 16T671 x 2

Low degree siblings

16T671 x 2, 16T692 x 2, 32T3265 x 2, 32T3267 x 4, 32T3268 x 4, 32T3269, 32T3333, 32T3334, 32T3335, 32T7344
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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$2$	$2$	$(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,32)(30,31)$
	$ 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$4$	$4$	$(17,26,19,27)(18,25,20,28)(21,29,23,32)(22,30,24,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$8$	$2$	$( 5,14)( 6,13)( 7,16)( 8,15)( 9,12)(10,11)(17,18)(19,20)(21,22)(23,24)(25,26) (27,28)(29,30)(31,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$8$	$2$	$( 5,14)( 6,13)( 7,16)( 8,15)( 9,12)(10,11)(17,20)(18,19)(21,24)(22,23)(25,27) (26,28)(29,31)(30,32)$
	$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$16$	$4$	$( 5,14)( 6,13)( 7,16)( 8,15)( 9,12)(10,11)(17,25,19,28)(18,26,20,27) (21,30,23,31)(22,29,24,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$16$	$2$	$( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,11)(10,12)(17,18)(19,20)(21,30) (22,29)(23,31)(24,32)(25,27)(26,28)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,23) (22,24)(25,28)(26,27)(29,32)(30,31)$
	$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$4$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)(17,26,19,27) (18,25,20,28)(21,29,23,32)(22,30,24,31)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$16$	$4$	$( 1, 5, 3, 7)( 2, 6, 4, 8)( 9,16,12,14)(10,15,11,13)(17,21,19,23)(18,22,20,24) (25,31,28,30)(26,32,27,29)$
	$ 8, 8, 4, 4, 4, 4 $	$16$	$8$	$( 1, 5, 9,14, 3, 7,12,16)( 2, 6,10,13, 4, 8,11,15)(17,22,19,24)(18,21,20,23) (25,32,28,29)(26,31,27,30)$
	$ 8, 8, 4, 4, 4, 4 $	$16$	$8$	$( 1, 6, 3, 8)( 2, 5, 4, 7)( 9,15,12,13)(10,16,11,14)(17,23,26,32,19,21,27,29) (18,24,25,31,20,22,28,30)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1, 6, 9,13, 3, 8,12,15)( 2, 5,10,14, 4, 7,11,16)(17,22,26,30,19,24,27,31) (18,21,25,29,20,23,28,32)$
	$ 8, 8, 8, 8 $	$8$	$8$	$( 1, 6, 9,13, 3, 8,12,15)( 2, 5,10,14, 4, 7,11,16)(17,24,26,31,19,22,27,30) (18,23,25,32,20,21,28,29)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1, 8, 9,15, 3, 6,12,13)( 2, 7,10,16, 4, 5,11,14)(17,24,26,31,19,22,27,30) (18,23,25,32,20,21,28,29)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1, 9, 3,12)( 2,10, 4,11)( 5,14, 7,16)( 6,13, 8,15)(17,26,19,27)(18,25,20,28) (21,29,23,32)(22,30,24,31)$
	$ 16, 16 $	$16$	$16$	$( 1,17, 6,22, 9,26,13,30, 3,19, 8,24,12,27,15,31)( 2,18, 5,21,10,25,14,29, 4, 20, 7,23,11,28,16,32)$
	$ 16, 16 $	$16$	$16$	$( 1,17, 8,24, 9,26,15,31, 3,19, 6,22,12,27,13,30)( 2,18, 7,23,10,25,16,32, 4, 20, 5,21,11,28,14,29)$
	$ 8, 8, 8, 8 $	$32$	$8$	$( 1,17, 5,29, 3,19, 7,32)( 2,18, 6,30, 4,20, 8,31)( 9,27,16,23,12,26,14,21) (10,28,15,24,11,25,13,22)$
	$ 16, 16 $	$16$	$16$	$( 1,21,13,25,12,29, 6,20, 3,23,15,28, 9,32, 8,18)( 2,22,14,26,11,30, 5,19, 4, 24,16,27,10,31, 7,17)$
	$ 16, 16 $	$16$	$16$	$( 1,21,15,28,12,29, 8,18, 3,23,13,25, 9,32, 6,20)( 2,22,16,27,11,30, 7,17, 4, 24,14,26,10,31, 5,19)$
	$ 8, 8, 8, 8 $	$32$	$8$	$( 1,21,14,19, 3,23,16,17)( 2,22,13,20, 4,24,15,18)( 5,26,12,32, 7,27, 9,29) ( 6,25,11,31, 8,28,10,30)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$256=2^{8}$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	$6$
Label:	256.505	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	8A1	8A-1	8B	8C1	8C-1	8D1	8D-1	16A1	16A-1	16A3	16A-3
	Size	1	1	2	8	8	16	4	4	4	16	16	4	4	8	16	16	32	32	16	16	16	16
	2 P	1A	1A	1A	1A	1A	1A	2B	2A	2B	2A	2B	4A	4A	4A	4B	4B	4D	4D	8A1	8A-1	8A1	8A-1
	Type
256.505.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
256.505.1b	R	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$
256.505.1c	R	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
256.505.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
256.505.1e1	C	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- i$	$i$	$- i$	$i$	$i$	$- i$
256.505.1e2	C	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$i$	$- i$	$i$	$- i$	$- i$	$i$
256.505.1f1	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- i$	$i$	$i$	$- i$	$- i$	$i$
256.505.1f2	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$i$	$- i$	$- i$	$i$	$i$	$- i$
256.505.2a	R	$2$	$2$	$2$	$0$	$0$	$- 2$	$2$	$2$	$2$	$- 2$	$0$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.505.2b	R	$2$	$2$	$2$	$0$	$0$	$- 2$	$2$	$2$	$2$	$2$	$0$	$- 2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.505.4a	R	$4$	$4$	$4$	$0$	$0$	$0$	$4$	$- 4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.505.4b	R	$4$	$4$	$4$	$- 2$	$- 2$	$0$	$- 4$	$0$	$0$	$0$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.505.4c	R	$4$	$4$	$4$	$2$	$2$	$0$	$- 4$	$0$	$0$	$0$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.505.4d1	C	$4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$2$	$0$	$0$	$0$	$0$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$
256.505.4d2	C	$4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$2$	$0$	$0$	$0$	$0$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$
256.505.4e1	R	$4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$2$	$2$	$- 2$	$0$	$0$	$0$	$0$	$- ζ_{8}^{- 1} - ζ_{8}$	$- ζ_{8}^{- 1} - ζ_{8}$	$ζ_{8}^{- 1} + ζ_{8}$	$ζ_{8}^{- 1} + ζ_{8}$
256.505.4e2	R	$4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$2$	$2$	$- 2$	$0$	$0$	$0$	$0$	$ζ_{8}^{- 1} + ζ_{8}$	$ζ_{8}^{- 1} + ζ_{8}$	$- ζ_{8}^{- 1} - ζ_{8}$	$- ζ_{8}^{- 1} - ζ_{8}$
256.505.4f1	C	$4$	$- 4$	$0$	$- 2$	$2$	$0$	$0$	$- 2$	$2$	$0$	$0$	$- 2 ζ_{8} - 2 ζ_{8}^{3}$	$2 ζ_{8} + 2 ζ_{8}^{3}$	$0$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$0$	$0$	$0$	$0$	$0$	$0$
256.505.4f2	C	$4$	$- 4$	$0$	$- 2$	$2$	$0$	$0$	$- 2$	$2$	$0$	$0$	$2 ζ_{8} + 2 ζ_{8}^{3}$	$- 2 ζ_{8} - 2 ζ_{8}^{3}$	$0$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$0$	$0$	$0$	$0$	$0$	$0$
256.505.4g1	C	$4$	$- 4$	$0$	$2$	$- 2$	$0$	$0$	$- 2$	$2$	$0$	$0$	$- 2 ζ_{8} - 2 ζ_{8}^{3}$	$2 ζ_{8} + 2 ζ_{8}^{3}$	$0$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$0$	$0$	$0$	$0$	$0$	$0$
256.505.4g2	C	$4$	$- 4$	$0$	$2$	$- 2$	$0$	$0$	$- 2$	$2$	$0$	$0$	$2 ζ_{8} + 2 ζ_{8}^{3}$	$- 2 ζ_{8} - 2 ζ_{8}^{3}$	$0$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$0$	$0$	$0$	$0$	$0$	$0$
256.505.8a	R	$8$	$- 8$	$0$	$0$	$0$	$0$	$0$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

magma: CharacterTable(G);

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Group action invariants

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Group invariants

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	32T3266
Degree	$32$
Order	$256$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$D_4^2.C_4$