LMFDB - Galois group: 32T3482

Show commands: Magma

magma: G := TransitiveGroup(32, 3482);

Group action invariants

Degree $n$:	$32$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$3482$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_2^4.C_2^4$
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:	(1,17,2,18)(3,20,4,19)(5,24,6,23)(7,21,8,22)(9,31,10,32)(11,30,12,29)(13,25,14,26)(15,28,16,27), (1,4)(2,3)(9,13,12,16)(10,14,11,15)(17,24,19,22)(18,23,20,21)(25,26)(27,28)(29,32)(30,31), (1,27)(2,28)(3,26)(4,25)(5,30)(6,29)(7,31)(8,32)(9,24)(10,23)(11,21)(12,22)(13,18)(14,17)(15,19)(16,20), (1,21,4,23)(2,22,3,24)(5,20,8,18)(6,19,7,17)(9,28,12,26)(10,27,11,25)(13,30,16,32)(14,29,15,31)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 15
$4$: $C_4$ x 8, $C_2^2$ x 35
$8$: $D_{4}$ x 8, $C_4\times C_2$ x 28, $C_2^3$ x 15
$16$: $D_4\times C_2$ x 12, $C_2^2:C_4$ x 16, $C_4\times C_2^2$ x 14, $C_2^4$
$32$: $Q_8:C_2^2$ x 4, $C_2 \times (C_2^2:C_4)$ x 12, $C_2^2 \times D_4$ x 2, 32T34
$64$: 16T68 x 2, 16T87 x 4, 32T262
$128$: 16T199 x 2, 32T992

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 15
$4$:	$C_4$ x 8, $C_2^2$ x 35
$8$:	$D_{4}$ x 8, $C_4\times C_2$ x 28, $C_2^3$ x 15
$16$:	$D_4\times C_2$ x 12, $C_2^2:C_4$ x 16, $C_4\times C_2^2$ x 14, $C_2^4$
$32$:	$Q_8:C_2^2$ x 4, $C_2 \times (C_2^2:C_4)$ x 12, $C_2^2 \times D_4$ x 2, 32T34
$64$:	16T68 x 2, 16T87 x 4, 32T262
$128$:	16T199 x 2, 32T992

Subfields

Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 8: $D_4\times C_2$, $Q_8:C_2^2$ x 2
Degree 16: 16T87, 16T199 x 2

Low degree siblings

32T3482 x 7, 32T3592 x 8, 32T3594 x 8, 32T3612 x 8
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, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $	$4$	$2$	$( 9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,27)(26,28)(29,31) (30,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $	$4$	$2$	$( 9,11)(10,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,27)(26,28)(29,31) (30,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$2$	$2$	$( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)$
	$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$8$	$4$	$( 5, 8)( 6, 7)( 9,13,12,16)(10,14,11,15)(17,24,19,22)(18,23,20,21)(25,28) (26,27)(29,30)(31,32)$
	$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$8$	$4$	$( 5, 8)( 6, 7)( 9,14,12,15)(10,13,11,16)(17,21,19,23)(18,22,20,24)(25,26) (27,28)(29,32)(30,31)$
	$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$8$	$4$	$( 5, 8)( 6, 7)( 9,15,12,14)(10,16,11,13)(17,23,19,21)(18,24,20,22)(25,26) (27,28)(29,32)(30,31)$
	$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$8$	$4$	$( 5, 8)( 6, 7)( 9,16,12,13)(10,15,11,14)(17,22,19,24)(18,21,20,23)(25,28) (26,27)(29,30)(31,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24) (22,23)(25,26)(27,28)(29,30)(31,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23) (22,24)(25,27)(26,28)(29,31)(30,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23) (20,24)(25,30)(26,29)(27,32)(28,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,14)(10,13)(11,16)(12,15)(17,24)(18,23)(19,22) (20,21)(25,32)(26,31)(27,30)(28,29)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,15)(10,16)(11,13)(12,14)(17,22)(18,21)(19,24) (20,23)(25,32)(26,31)(27,30)(28,29)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,16)(10,15)(11,14)(12,13)(17,23)(18,24)(19,21) (20,22)(25,30)(26,29)(27,32)(28,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,13)(10,14)(11,15)(12,16)(17,23)(18,24)(19,21) (20,22)(25,31)(26,32)(27,29)(28,30)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,16)(10,15)(11,14)(12,13)(17,21)(18,22)(19,23) (20,24)(25,31)(26,32)(27,29)(28,30)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$8$	$2$	$( 1, 9)( 2,10)( 3,11)( 4,12)( 5,14)( 6,13)( 7,16)( 8,15)(17,30)(18,29)(19,32) (20,31)(21,26)(22,25)(23,28)(24,27)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1, 9, 2,10)( 3,11, 4,12)( 5,14, 6,13)( 7,16, 8,15)(17,31,18,32)(19,29,20,30) (21,27,22,28)(23,25,24,26)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1, 9, 3,11)( 2,10, 4,12)( 5,14, 7,16)( 6,13, 8,15)(17,29,20,32)(18,30,19,31) (21,25,24,28)(22,26,23,27)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1, 9, 4,12)( 2,10, 3,11)( 5,14, 8,15)( 6,13, 7,16)(17,32,19,30)(18,31,20,29) (21,28,23,26)(22,27,24,25)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1, 9, 5,15)( 2,10, 6,16)( 3,11, 7,13)( 4,12, 8,14)(17,26,22,31)(18,25,21,32) (19,28,24,29)(20,27,23,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1, 9, 6,16)( 2,10, 5,15)( 3,11, 8,14)( 4,12, 7,13)(17,27,21,29)(18,28,22,30) (19,25,23,31)(20,26,24,32)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1, 9, 7,13)( 2,10, 8,14)( 3,11, 5,15)( 4,12, 6,16)(17,25,23,29)(18,26,24,30) (19,27,21,31)(20,28,22,32)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1, 9, 8,14)( 2,10, 7,13)( 3,11, 6,16)( 4,12, 5,15)(17,28,24,31)(18,27,23,32) (19,26,22,29)(20,25,21,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1,17, 6,21)( 2,18, 5,22)( 3,20, 8,24)( 4,19, 7,23)( 9,25,16,31)(10,26,15,32) (11,28,14,30)(12,27,13,29)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1,17, 8,24)( 2,18, 7,23)( 3,20, 6,21)( 4,19, 5,22)( 9,26,14,29)(10,25,13,30) (11,27,16,32)(12,28,15,31)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1,17, 7,23)( 2,18, 8,24)( 3,20, 5,22)( 4,19, 6,21)( 9,27,13,31)(10,28,14,32) (11,26,15,30)(12,25,16,29)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1,17, 5,22)( 2,18, 6,21)( 3,20, 7,23)( 4,19, 8,24)( 9,28,15,29)(10,27,16,30) (11,25,13,32)(12,26,14,31)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1,17, 3,20)( 2,18, 4,19)( 5,24, 7,21)( 6,23, 8,22)( 9,29,11,32)(10,30,12,31) (13,27,15,26)(14,28,16,25)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$8$	$2$	$( 1,17)( 2,18)( 3,20)( 4,19)( 5,24)( 6,23)( 7,21)( 8,22)( 9,30)(10,29)(11,31) (12,32)(13,28)(14,27)(15,25)(16,26)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1,17, 2,18)( 3,20, 4,19)( 5,24, 6,23)( 7,21, 8,22)( 9,31,10,32)(11,30,12,29) (13,25,14,26)(15,28,16,27)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1,17, 4,19)( 2,18, 3,20)( 5,24, 8,22)( 6,23, 7,21)( 9,32,12,30)(10,31,11,29) (13,26,16,28)(14,25,15,27)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1,25, 2,26)( 3,28, 4,27)( 5,30, 6,29)( 7,31, 8,32)( 9,17,11,20)(10,18,12,19) (13,21,15,24)(14,22,16,23)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1,25, 3,28)( 2,26, 4,27)( 5,30, 7,31)( 6,29, 8,32)( 9,18,10,17)(11,19,12,20) (13,22,14,21)(15,23,16,24)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1,25, 2,26)( 3,28, 4,27)( 5,30, 6,29)( 7,31, 8,32)( 9,19,11,18)(10,20,12,17) (13,23,15,22)(14,24,16,21)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$8$	$4$	$( 1,25, 3,28)( 2,26, 4,27)( 5,30, 7,31)( 6,29, 8,32)( 9,20,10,19)(11,17,12,18) (13,24,14,23)(15,21,16,22)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,25, 4,27)( 2,26, 3,28)( 5,32, 8,30)( 6,31, 7,29)( 9,21,12,23)(10,22,11,24) (13,19,16,17)(14,20,15,18)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1,25)( 2,26)( 3,28)( 4,27)( 5,32)( 6,31)( 7,29)( 8,30)( 9,22)(10,21)(11,23) (12,24)(13,20)(14,19)(15,17)(16,18)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,25, 4,27)( 2,26, 3,28)( 5,32, 8,30)( 6,31, 7,29)( 9,23,12,21)(10,24,11,22) (13,17,16,19)(14,18,15,20)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1,25)( 2,26)( 3,28)( 4,27)( 5,32)( 6,31)( 7,29)( 8,30)( 9,24)(10,23)(11,21) (12,22)(13,18)(14,17)(15,19)(16,20)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,29, 4,31)( 2,30, 3,32)( 5,28, 8,26)( 6,27, 7,25)( 9,17,12,19)(10,18,11,20) (13,23,16,21)(14,24,15,22)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1,29)( 2,30)( 3,32)( 4,31)( 5,28)( 6,27)( 7,25)( 8,26)( 9,18)(10,17)(11,19) (12,20)(13,24)(14,23)(15,21)(16,22)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,29, 4,31)( 2,30, 3,32)( 5,28, 8,26)( 6,27, 7,25)( 9,19,12,17)(10,20,11,18) (13,21,16,23)(14,22,15,24)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1,29)( 2,30)( 3,32)( 4,31)( 5,28)( 6,27)( 7,25)( 8,26)( 9,20)(10,19)(11,17) (12,18)(13,22)(14,21)(15,23)(16,24)$

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:	$3$
Label:	256.14737	magma: IdentifyGroup(G);
Character table:	46 x 46 character table

magma: CharacterTable(G);

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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	32T3482
Degree	$32$
Order	$256$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$C_2^4.C_2^4$