LMFDB - Galois group: 32T430

Show commands: Magma

magma: G := TransitiveGroup(32, 430);

Group action invariants

Degree $n$:	$32$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$430$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$\OD_{16}.C_2^3$
Parity:	$1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$8$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,6)(2,5)(3,7)(4,8)(9,16)(10,15)(11,13)(12,14)(17,24)(18,23)(19,22)(20,21)(25,31)(26,32)(27,29)(28,30), (1,13)(2,14)(3,15)(4,16)(5,10)(6,9)(7,12)(8,11)(17,31)(18,32)(19,30)(20,29)(21,26)(22,25)(23,27)(24,28), (1,20,10,28,4,18,12,25)(2,19,9,27,3,17,11,26)(5,24,16,32,7,22,13,29)(6,23,15,31,8,21,14,30), (1,25,10,18,4,28,12,20)(2,26,9,17,3,27,11,19)(5,32,16,24,7,29,13,22)(6,31,15,23,8,30,14,21)	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, $C_2^4$
$32$: $C_2 \times (C_2^2:C_4)$ x 12, $C_2^2 \times D_4$ x 2, 32T34
$64$: 32T262

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, $C_2^4$
$32$:	$C_2 \times (C_2^2:C_4)$ x 12, $C_2^2 \times D_4$ x 2, 32T34
$64$:	32T262

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, $C_2^3$
Degree 16: $C_4\times C_2^2$, 16T200 x 2

Low degree siblings

16T200 x 6, 32T428 x 3, 32T429 x 12, 32T430 x 5
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 $	$2$	$2$	$(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,32)(30,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$4$	$2$	$( 5, 7)( 6, 8)( 9,11)(10,12)(21,23)(22,24)(25,28)(26,27)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,15)(14,16)(17,18)(19,20)(21,22) (23,24)(25,27)(26,28)(29,31)(30,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 2)( 3, 4)( 5, 8)( 6, 7)( 9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,24) (22,23)(25,26)(27,28)(29,31)(30,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 2)( 3, 4)( 5, 8)( 6, 7)( 9,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,22) (23,24)(25,27)(26,28)(29,30)(31,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,23) (22,24)(25,28)(26,27)(29,32)(30,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,14)(10,13)(11,15)(12,16)(17,21)(18,22)(19,23) (20,24)(25,32)(26,31)(27,30)(28,29)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,15,11,14)(10,16,12,13)(17,21,19,23)(18,22,20,24) (25,29,28,32)(26,30,27,31)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,15,11,14)(10,16,12,13)(17,23,19,21)(18,24,20,22) (25,32,28,29)(26,31,27,30)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,16)(10,15)(11,13)(12,14)(17,22)(18,21)(19,24) (20,23)(25,30)(26,29)(27,32)(28,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,16)(10,15)(11,13)(12,14)(17,24)(18,23)(19,22) (20,21)(25,31)(26,32)(27,29)(28,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1, 6, 4, 8)( 2, 5, 3, 7)( 9,13,11,16)(10,14,12,15)(17,22,19,24)(18,21,20,23) (25,31,28,30)(26,32,27,29)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1, 9, 4,11)( 2,10, 3,12)( 5,14, 7,15)( 6,13, 8,16)(17,25,19,28)(18,26,20,27) (21,32,23,29)(22,31,24,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1, 9, 4,11)( 2,10, 3,12)( 5,14, 7,15)( 6,13, 8,16)(17,28,19,25)(18,27,20,26) (21,29,23,32)(22,30,24,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1, 9)( 2,10)( 3,12)( 4,11)( 5,15)( 6,16)( 7,14)( 8,13)(17,25)(18,26)(19,28) (20,27)(21,29)(22,30)(23,32)(24,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1,10)( 2, 9)( 3,11)( 4,12)( 5,13)( 6,14)( 7,16)( 8,15)(17,26)(18,25)(19,27) (20,28)(21,31)(22,32)(23,30)(24,29)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,10, 4,12)( 2, 9, 3,11)( 5,16, 7,13)( 6,15, 8,14)(17,26,19,27)(18,25,20,28) (21,30,23,31)(22,29,24,32)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,10, 4,12)( 2, 9, 3,11)( 5,16, 7,13)( 6,15, 8,14)(17,27,19,26)(18,28,20,25) (21,31,23,30)(22,32,24,29)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1,13)( 2,14)( 3,15)( 4,16)( 5,10)( 6, 9)( 7,12)( 8,11)(17,30)(18,29)(19,31) (20,32)(21,27)(22,28)(23,26)(24,25)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1,13)( 2,14)( 3,15)( 4,16)( 5,10)( 6, 9)( 7,12)( 8,11)(17,31)(18,32)(19,30) (20,29)(21,26)(22,25)(23,27)(24,28)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1,13)( 2,14)( 3,15)( 4,16)( 5,12)( 6,11)( 7,10)( 8, 9)(17,30)(18,29)(19,31) (20,32)(21,26)(22,25)(23,27)(24,28)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,14, 4,15)( 2,13, 3,16)( 5, 9, 7,11)( 6,10, 8,12)(17,29,19,32)(18,30,20,31) (21,28,23,25)(22,27,24,26)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,14, 4,15)( 2,13, 3,16)( 5,11, 7, 9)( 6,12, 8,10)(17,29,19,32)(18,30,20,31) (21,25,23,28)(22,26,24,27)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,14, 4,15)( 2,13, 3,16)( 5,11, 7, 9)( 6,12, 8,10)(17,32,19,29)(18,31,20,30) (21,28,23,25)(22,27,24,26)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,17,10,27, 4,19,12,26)( 2,18, 9,28, 3,20,11,25)( 5,21,16,31, 7,23,13,30) ( 6,22,15,32, 8,24,14,29)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,17,10,26, 4,19,12,27)( 2,18, 9,25, 3,20,11,28)( 5,23,16,31, 7,21,13,30) ( 6,24,15,32, 8,22,14,29)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,18,10,25, 4,20,12,28)( 2,17, 9,26, 3,19,11,27)( 5,22,16,29, 7,24,13,32) ( 6,21,15,30, 8,23,14,31)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,18,10,28, 4,20,12,25)( 2,17, 9,27, 3,19,11,26)( 5,24,16,29, 7,22,13,32) ( 6,23,15,30, 8,21,14,31)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,21,12,31, 4,23,10,30)( 2,22,11,32, 3,24, 9,29)( 5,17,13,27, 7,19,16,26) ( 6,18,14,28, 8,20,15,25)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,21,10,31, 4,23,12,30)( 2,22, 9,32, 3,24,11,29)( 5,19,16,26, 7,17,13,27) ( 6,20,15,25, 8,18,14,28)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,22,12,29, 4,24,10,32)( 2,21,11,30, 3,23, 9,31)( 5,18,13,25, 7,20,16,28) ( 6,17,14,26, 8,19,15,27)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,22,10,29, 4,24,12,32)( 2,21, 9,30, 3,23,11,31)( 5,20,16,28, 7,18,13,25) ( 6,19,15,27, 8,17,14,26)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,25,10,20, 4,28,12,18)( 2,26, 9,19, 3,27,11,17)( 5,29,16,24, 7,32,13,22) ( 6,30,15,23, 8,31,14,21)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,25,12,20, 4,28,10,18)( 2,26,11,19, 3,27, 9,17)( 5,32,13,22, 7,29,16,24) ( 6,31,14,21, 8,30,15,23)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,26,12,19, 4,27,10,17)( 2,25,11,20, 3,28, 9,18)( 5,30,13,23, 7,31,16,21) ( 6,29,14,24, 8,32,15,22)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,26,10,19, 4,27,12,17)( 2,25, 9,20, 3,28,11,18)( 5,31,16,21, 7,30,13,23) ( 6,32,15,22, 8,29,14,24)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,29,12,24, 4,32,10,22)( 2,30,11,23, 3,31, 9,21)( 5,25,13,20, 7,28,16,18) ( 6,26,14,19, 8,27,15,17)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,29,12,22, 4,32,10,24)( 2,30,11,21, 3,31, 9,23)( 5,28,13,20, 7,25,16,18) ( 6,27,14,19, 8,26,15,17)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,30,10,23, 4,31,12,21)( 2,29, 9,24, 3,32,11,22)( 5,26,16,19, 7,27,13,17) ( 6,25,15,20, 8,28,14,18)$
	$ 8, 8, 8, 8 $	$4$	$8$	$( 1,30,10,21, 4,31,12,23)( 2,29, 9,22, 3,32,11,24)( 5,27,16,19, 7,26,13,17) ( 6,28,15,20, 8,25,14,18)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$128=2^{7}$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	$3$
Label:	128.1620	magma: IdentifyGroup(G);
Character table:	41 x 41 character table

magma: CharacterTable(G);

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Group action invariants

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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	32T430
Degree	$32$
Order	$128$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$\OD_{16}.C_2^3$