LMFDB - Galois group: 32T3

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magma: G := TransitiveGroup(32, 3);

Group action invariants

Degree $n$:	$32$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$3$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_4^2:C_2$
Parity:	$1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$32$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,21,3,24)(2,22,4,23)(5,17,8,20)(6,18,7,19)(9,31,11,30)(10,32,12,29)(13,25,16,28)(14,26,15,27), (1,18)(2,17)(3,19)(4,20)(5,7)(6,8)(9,28)(10,27)(11,25)(12,26)(13,15)(14,16)(21,23)(22,24)(29,31)(30,32), (1,14,17,31)(2,13,18,32)(3,15,20,30)(4,16,19,29)(5,9,24,27)(6,10,23,28)(7,12,22,25)(8,11,21,26)	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
$8$: $C_4\times C_2$ x 6, $C_2^3$
$16$: $Q_8:C_2$ x 2, $C_4\times C_2^2$

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
$8$:	$C_4\times C_2$ x 6, $C_2^3$
$16$:	$Q_8:C_2$ x 2, $C_4\times C_2^2$

Subfields

Degree 2: $C_2$ x 7
Degree 4: $C_4$ x 4, $C_2^2$ x 7
Degree 8: $C_4\times C_2$ x 6, $C_2^3$, $Q_8:C_2$ x 6
Degree 16: $C_4\times C_2^2$, $Q_8 : C_2$ x 2, $C_4^2:C_2$ x 2

Low degree siblings

16T17 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	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, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 2)( 3, 4)( 5,22)( 6,21)( 7,24)( 8,23)( 9,10)(11,12)(13,30)(14,29)(15,32) (16,31)(17,18)(19,20)(25,26)(27,28)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 4)( 2, 3)( 5,23)( 6,24)( 7,21)( 8,22)( 9,12)(10,11)(13,31)(14,32)(15,29) (16,30)(17,19)(18,20)(25,27)(26,28)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1, 5, 3, 8)( 2, 6, 4, 7)( 9,15,11,14)(10,16,12,13)(17,24,20,21)(18,23,19,22) (25,32,28,29)(26,31,27,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1, 6,17,23)( 2, 5,18,24)( 3, 7,20,22)( 4, 8,19,21)( 9,16,27,29)(10,15,28,30) (11,13,26,32)(12,14,25,31)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1, 7,17,22)( 2, 8,18,21)( 3, 6,20,23)( 4, 5,19,24)( 9,13,27,32)(10,14,28,31) (11,16,26,29)(12,15,25,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1, 8, 3, 5)( 2, 7, 4, 6)( 9,14,11,15)(10,13,12,16)(17,21,20,24)(18,22,19,23) (25,29,28,32)(26,30,27,31)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1, 9,20,26)( 2,10,19,25)( 3,11,17,27)( 4,12,18,28)( 5,15,21,31)( 6,16,22,32) ( 7,13,23,29)( 8,14,24,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,10,20,25)( 2, 9,19,26)( 3,12,17,28)( 4,11,18,27)( 5,32,21,16)( 6,31,22,15) ( 7,30,23,14)( 8,29,24,13)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,11,20,27)( 2,12,19,28)( 3, 9,17,26)( 4,10,18,25)( 5,14,21,30)( 6,13,22,29) ( 7,16,23,32)( 8,15,24,31)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,12,20,28)( 2,11,19,27)( 3,10,17,25)( 4, 9,18,26)( 5,29,21,13)( 6,30,22,14) ( 7,31,23,15)( 8,32,24,16)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,13, 3,16)( 2,14, 4,15)( 5,25, 8,28)( 6,26, 7,27)( 9,23,11,22)(10,24,12,21) (17,32,20,29)(18,31,19,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,14,17,31)( 2,13,18,32)( 3,15,20,30)( 4,16,19,29)( 5, 9,24,27)( 6,10,23,28) ( 7,12,22,25)( 8,11,21,26)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,15,17,30)( 2,16,18,29)( 3,14,20,31)( 4,13,19,32)( 5,11,24,26)( 6,12,23,25) ( 7,10,22,28)( 8, 9,21,27)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,16, 3,13)( 2,15, 4,14)( 5,28, 8,25)( 6,27, 7,26)( 9,22,11,23)(10,21,12,24) (17,29,20,32)(18,30,19,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,17)( 2,18)( 3,20)( 4,19)( 5,24)( 6,23)( 7,22)( 8,21)( 9,27)(10,28)(11,26) (12,25)(13,32)(14,31)(15,30)(16,29)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,20)( 2,19)( 3,17)( 4,18)( 5,21)( 6,22)( 7,23)( 8,24)( 9,26)(10,25)(11,27) (12,28)(13,29)(14,30)(15,31)(16,32)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,26,20, 9)( 2,25,19,10)( 3,27,17,11)( 4,28,18,12)( 5,31,21,15)( 6,32,22,16) ( 7,29,23,13)( 8,30,24,14)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,27,20,11)( 2,28,19,12)( 3,26,17, 9)( 4,25,18,10)( 5,30,21,14)( 6,29,22,13) ( 7,32,23,16)( 8,31,24,15)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$32=2^{5}$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	$2$
Label:	32.24	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	2C	2D	2E	4A1	4A-1	4B1	4B-1	4C	4D	4E1	4E-1	4F1	4F-1	4G1	4G-1	4H1	4H-1
	Size	1	1	1	1	2	2	1	1	1	1	2	2	2	2	2	2	2	2	2	2
	2 P	1A	1A	1A	1A	1A	1A	2A	2A	2A	2A	2C	2B	2B	2C	2C	2B	2B	2C	2A	2A
	Type
32.24.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.24.1b	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$
32.24.1c	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
32.24.1d	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$
32.24.1e	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$
32.24.1f	R	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$
32.24.1g	R	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$
32.24.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
32.24.1i1	C	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- i$	$i$	$- i$	$i$	$i$	$- i$	$i$	$- i$
32.24.1i2	C	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$i$	$- i$	$i$	$- i$	$- i$	$i$	$- i$	$i$
32.24.1j1	C	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- i$	$i$	$i$	$- i$	$- i$	$i$	$i$	$- i$
32.24.1j2	C	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$i$	$- i$	$- i$	$i$	$i$	$- i$	$- i$	$i$
32.24.1k1	C	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$- i$	$i$	$i$	$- i$	$i$	$- i$	$- i$	$i$
32.24.1k2	C	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$i$	$- i$	$- i$	$i$	$- i$	$i$	$i$	$- i$
32.24.1l1	C	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- i$	$i$	$- i$	$i$	$- i$	$i$	$- i$	$i$
32.24.1l2	C	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$i$	$- i$	$i$	$- i$	$i$	$- i$	$i$	$- i$
32.24.2a1	C	$2$	$- 2$	$- 2$	$2$	$0$	$0$	$- 2 i$	$2 i$	$- 2 i$	$2 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.24.2a2	C	$2$	$- 2$	$- 2$	$2$	$0$	$0$	$2 i$	$- 2 i$	$2 i$	$- 2 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.24.2b1	C	$2$	$- 2$	$2$	$- 2$	$0$	$0$	$- 2 i$	$2 i$	$2 i$	$- 2 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.24.2b2	C	$2$	$- 2$	$2$	$- 2$	$0$	$0$	$2 i$	$- 2 i$	$- 2 i$	$2 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

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	32T3
Degree	$32$
Order	$32$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$C_4^2:C_2$