LMFDB - Galois group: 32T2652

Show commands: Magma

magma: G := TransitiveGroup(32, 2652);

Group action invariants

Degree $n$:	$32$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$2652$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$(C_4\times C_8).D_4$
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:	(1,24,4,21)(2,23,3,22)(5,19,8,18)(6,20,7,17)(9,31)(10,32)(11,30)(12,29)(13,27)(14,28)(15,25)(16,26), (1,18,8,24,4,19,5,21)(2,17,7,23,3,20,6,22)(9,27,14,32,11,26,15,29)(10,28,13,31,12,25,16,30), (1,10,23,27,5,13,17,32,4,12,22,26,8,16,20,29)(2,9,24,28,6,14,18,31,3,11,21,25,7,15,19,30)	magma: Generators(G);

Low degree resolvents

|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, $C_4\times C_2^2$
$32$: $(C_8:C_2):C_2$ x 2, $C_2 \times (C_2^2:C_4)$
$64$: 16T72, 16T106 x 2
$128$: 32T1510

Resolvents shown for degrees $\leq 47$

\|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, $C_4\times C_2^2$
$32$:	$(C_8:C_2):C_2$ x 2, $C_2 \times (C_2^2:C_4)$
$64$:	16T72, 16T106 x 2
$128$:	32T1510

Subfields

Degree 2: $C_2$
Degree 4: $C_4$, $D_{4}$ x 2
Degree 8: $C_2^2:C_4$, $(C_8:C_2):C_2$
Degree 16: 16T41, 16T534 x 2

Low degree siblings

16T534 x 4, 32T2650 x 2, 32T2651 x 4, 32T2652 x 3, 32T2653 x 4, 32T6936 x 2
Siblings are shown with degree $\leq 47$

A number field with this Galois group has 3 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 $	$4$	$2$	$(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$2$	$2$	$( 9,11)(10,12)(13,16)(14,15)(25,28)(26,27)(29,32)(30,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$16$	$2$	$( 5, 8)( 6, 7)( 9,10)(11,12)(13,15)(14,16)(17,22)(18,21)(19,24)(20,23)(25,29) (26,30)(27,31)(28,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$16$	$2$	$( 5, 8)( 6, 7)( 9,10)(11,12)(13,15)(14,16)(17,23)(18,24)(19,21)(20,22)(25,32) (26,31)(27,30)(28,29)$
	$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$4$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,13,11,16)(10,14,12,15)(17,18)(19,20)(21,22) (23,24)(25,29,28,32)(26,30,27,31)$
	$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2 $	$8$	$4$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,13,11,16)(10,14,12,15)(17,19)(18,20)(21,23) (22,24)(25,32,28,29)(26,31,27,30)$
	$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$4$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,13,11,16)(10,14,12,15)(17,19)(18,20)(21,23) (22,24)(25,29,28,32)(26,30,27,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1, 5, 4, 8)( 2, 6, 3, 7)( 9,14,11,15)(10,13,12,16)(17,22,20,23)(18,21,19,24) (25,30,28,31)(26,29,27,32)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1, 5, 4, 8)( 2, 6, 3, 7)( 9,14,11,15)(10,13,12,16)(17,23,20,22)(18,24,19,21) (25,31,28,30)(26,32,27,29)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1, 5, 4, 8)( 2, 6, 3, 7)( 9,15,11,14)(10,16,12,13)(17,22,20,23)(18,21,19,24) (25,31,28,30)(26,32,27,29)$
	$ 16, 16 $	$16$	$16$	$( 1, 9,17,30, 8,15,23,25, 4,11,20,31, 5,14,22,28)( 2,10,18,29, 7,16,24,26, 3, 12,19,32, 6,13,21,27)$
	$ 16, 16 $	$16$	$16$	$( 1, 9,17,31, 8,15,23,28, 4,11,20,30, 5,14,22,25)( 2,10,18,32, 7,16,24,27, 3, 12,19,29, 6,13,21,26)$
	$ 16, 16 $	$16$	$16$	$( 1, 9,18,26, 5,15,21,32, 4,11,19,27, 8,14,24,29)( 2,10,17,25, 6,16,22,31, 3, 12,20,28, 7,13,23,30)$
	$ 16, 16 $	$16$	$16$	$( 1, 9,18,27, 5,15,21,29, 4,11,19,26, 8,14,24,32)( 2,10,17,28, 6,16,22,30, 3, 12,20,25, 7,13,23,31)$
	$ 8, 8, 8, 8 $	$8$	$8$	$( 1,17, 5,22, 4,20, 8,23)( 2,18, 6,21, 3,19, 7,24)( 9,30,14,28,11,31,15,25) (10,29,13,27,12,32,16,26)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,17, 8,23, 4,20, 5,22)( 2,18, 7,24, 3,19, 6,21)( 9,30,15,25,11,31,14,28) (10,29,16,26,12,32,13,27)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,17, 8,23, 4,20, 5,22)( 2,18, 7,24, 3,19, 6,21)( 9,31,15,28,11,30,14,25) (10,32,16,27,12,29,13,26)$
	$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2 $	$16$	$4$	$( 1,17)( 2,18)( 3,19)( 4,20)( 5,23)( 6,24)( 7,21)( 8,22)( 9,29,11,32) (10,30,12,31)(13,25,16,28)(14,26,15,27)$
	$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2 $	$16$	$4$	$( 1,17, 4,20)( 2,18, 3,19)( 5,23, 8,22)( 6,24, 7,21)( 9,29)(10,30)(11,32) (12,31)(13,25)(14,26)(15,27)(16,28)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,18, 5,21, 4,19, 8,24)( 2,17, 6,22, 3,20, 7,23)( 9,26,15,32,11,27,14,29) (10,25,16,31,12,28,13,30)$
	$ 8, 8, 8, 8 $	$8$	$8$	$( 1,18, 8,24, 4,19, 5,21)( 2,17, 7,23, 3,20, 6,22)( 9,26,14,29,11,27,15,32) (10,25,13,30,12,28,16,31)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,18, 5,21, 4,19, 8,24)( 2,17, 6,22, 3,20, 7,23)( 9,27,15,29,11,26,14,32) (10,28,16,30,12,25,13,31)$
	$ 16, 16 $	$16$	$16$	$( 1,25,22,14, 5,30,20,11, 4,28,23,15, 8,31,17, 9)( 2,26,21,13, 6,29,19,12, 3, 27,24,16, 7,32,18,10)$
	$ 16, 16 $	$16$	$16$	$( 1,25,23,14, 5,30,17,11, 4,28,22,15, 8,31,20, 9)( 2,26,24,13, 6,29,18,12, 3, 27,21,16, 7,32,19,10)$
	$ 16, 16 $	$16$	$16$	$( 1,25,19,16, 5,31,24,12, 4,28,18,13, 8,30,21,10)( 2,26,20,15, 6,32,23,11, 3, 27,17,14, 7,29,22, 9)$
	$ 16, 16 $	$16$	$16$	$( 1,25,18,16, 5,31,21,12, 4,28,19,13, 8,30,24,10)( 2,26,17,15, 6,32,22,11, 3, 27,20,14, 7,29,23, 9)$

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:	$4$
Label:	256.4903	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	8A1	8A-1	8B1	8B-1	8C	8D	16A1	16A-1	16B1	16B-1	16C1	16C-1	16D1	16D-1
	Size	1	1	2	4	16	16	2	2	4	4	4	8	16	16	4	4	4	4	8	8	16	16	16	16	16	16	16	16
	2 P	1A	1A	1A	1A	1A	1A	2A	2A	2A	2B	2B	2B	2B	2B	4A	4A	4B	4B	4A	4B	8B-1	8A1	8B-1	8A1	8A-1	8B1	8B1	8A-1
	Type
256.4903.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
256.4903.1b	R	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$
256.4903.1c	R	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$
256.4903.1d	R	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$
256.4903.1e	R	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
256.4903.1f	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$
256.4903.1g	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$
256.4903.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
256.4903.1i1	C	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- i$	$i$	$- i$	$i$	$- i$	$i$	$- i$	$i$
256.4903.1i2	C	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$i$	$- i$	$i$	$- i$	$i$	$- i$	$i$	$- i$
256.4903.1j1	C	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- i$	$i$	$i$	$- i$	$i$	$- i$	$- i$	$i$
256.4903.1j2	C	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$i$	$- i$	$- i$	$i$	$- i$	$i$	$i$	$- i$
256.4903.1k1	C	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- i$	$i$	$- i$	$i$	$i$	$- i$	$i$	$- i$
256.4903.1k2	C	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$i$	$- i$	$i$	$- i$	$- i$	$i$	$- i$	$i$
256.4903.1l1	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- i$	$i$	$i$	$- i$	$- i$	$i$	$i$	$- i$
256.4903.1l2	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$i$	$- i$	$- i$	$i$	$i$	$- i$	$- i$	$i$
256.4903.2a	R	$2$	$2$	$2$	$- 2$	$0$	$0$	$2$	$2$	$2$	$- 2$	$2$	$- 2$	$0$	$0$	$- 2$	$- 2$	$2$	$2$	$2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.4903.2b	R	$2$	$2$	$2$	$- 2$	$0$	$0$	$2$	$2$	$2$	$- 2$	$2$	$- 2$	$0$	$0$	$2$	$2$	$- 2$	$- 2$	$- 2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.4903.2c	R	$2$	$2$	$2$	$2$	$0$	$0$	$2$	$2$	$- 2$	$2$	$- 2$	$- 2$	$0$	$0$	$- 2$	$- 2$	$2$	$2$	$- 2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.4903.2d	R	$2$	$2$	$2$	$2$	$0$	$0$	$2$	$2$	$- 2$	$2$	$- 2$	$- 2$	$0$	$0$	$2$	$2$	$- 2$	$- 2$	$2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.4903.4a	R	$4$	$4$	$4$	$- 4$	$0$	$0$	$- 4$	$- 4$	$0$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.4903.4b	R	$4$	$4$	$4$	$4$	$0$	$0$	$- 4$	$- 4$	$0$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.4903.4c1	C	$4$	$4$	$- 4$	$0$	$0$	$0$	$- 4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$- 4 i$	$4 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.4903.4c2	C	$4$	$4$	$- 4$	$0$	$0$	$0$	$- 4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$4 i$	$- 4 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.4903.4d1	C	$4$	$4$	$- 4$	$0$	$0$	$0$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 4 i$	$4 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.4903.4d2	C	$4$	$4$	$- 4$	$0$	$0$	$0$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$4 i$	$- 4 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.4903.8a	R	$8$	$- 8$	$0$	$0$	$0$	$0$	$0$	$0$	$- 4$	$0$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.4903.8b	R	$8$	$- 8$	$0$	$0$	$0$	$0$	$0$	$0$	$4$	$0$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

magma: CharacterTable(G);

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Group action invariants

Low degree resolvents

Subfields

Low degree siblings

Conjugacy classes

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	32T2652
Degree	$32$
Order	$256$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$(C_4\times C_8).D_4$