LMFDB - Galois group: 32T1006

Show commands: Magma

magma: G := TransitiveGroup(32, 1006);

Group action invariants

Degree $n$:	$32$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$1006$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_2^3.C_2^4$
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:	(1,15,3,13)(2,16,4,14)(5,26,7,28)(6,25,8,27)(9,24,11,22)(10,23,12,21)(17,29,19,31)(18,30,20,32), (1,19,29,15)(2,20,30,16)(3,17,31,13)(4,18,32,14)(5,22,12,25)(6,21,11,26)(7,24,10,27)(8,23,9,28), (1,13,30,18)(2,14,29,17)(3,15,32,20)(4,16,31,19)(5,21,11,25)(6,22,12,26)(7,23,9,27)(8,24,10,28), (1,22)(2,21)(3,24)(4,23)(5,18)(6,17)(7,20)(8,19)(9,15)(10,16)(11,13)(12,14)(25,29)(26,30)(27,31)(28,32)	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
$8$: $C_4\times C_2$ x 28, $C_2^3$ x 15
$16$: $C_4\times C_2^2$ x 14, $C_2^4$
$32$: $Q_8:C_2^2$ x 6, 32T34
$64$: 16T68 x 3, 16T83 x 3, 16T98

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
$8$:	$C_4\times C_2$ x 28, $C_2^3$ x 15
$16$:	$C_4\times C_2^2$ x 14, $C_2^4$
$32$:	$Q_8:C_2^2$ x 6, 32T34
$64$:	16T68 x 3, 16T83 x 3, 16T98

Subfields

Degree 2: $C_2$ x 3
Degree 4: $C_4$ x 2, $C_2^2$
Degree 8: $C_4\times C_2$, $Q_8:C_2^2$ x 6
Degree 16: 16T68 x 3, 16T83 x 3, 16T98

Low degree siblings

32T1006 x 15
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 $	$4$	$2$	$( 5, 6)( 7, 8)( 9,10)(11,12)(13,19)(14,20)(15,17)(16,18)(21,27)(22,28)(23,25) (24,26)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $	$4$	$2$	$( 5, 9)( 6,10)( 7,11)( 8,12)(13,14)(15,16)(17,18)(19,20)(21,28)(22,27)(23,26) (24,25)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $	$4$	$2$	$( 5,10)( 6, 9)( 7,12)( 8,11)(13,20)(14,19)(15,18)(16,17)(21,22)(23,24)(25,26) (27,28)$
	$ 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 $	$1$	$2$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23) (22,24)(25,27)(26,28)(29,31)(30,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,17)(14,18)(15,19)(16,20)(21,25) (22,26)(23,27)(24,28)(29,31)(30,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1, 3)( 2, 4)( 5,11)( 6,12)( 7, 9)( 8,10)(13,16)(14,15)(17,20)(18,19)(21,26) (22,25)(23,28)(24,27)(29,31)(30,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1, 3)( 2, 4)( 5,12)( 6,11)( 7,10)( 8, 9)(13,18)(14,17)(15,20)(16,19)(21,24) (22,23)(25,28)(26,27)(29,31)(30,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1, 5, 3, 7)( 2, 6, 4, 8)( 9,30,11,32)(10,29,12,31)(13,28,15,26)(14,27,16,25) (17,23,19,21)(18,24,20,22)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1, 5, 4, 8)( 2, 6, 3, 7)( 9,29,12,32)(10,30,11,31)(13,21,16,24)(14,22,15,23) (17,26,20,27)(18,25,19,28)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1, 5,29,12)( 2, 6,30,11)( 3, 7,31,10)( 4, 8,32, 9)(13,22,17,25)(14,21,18,26) (15,24,19,27)(16,23,20,28)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1, 5,30,11)( 2, 6,29,12)( 3, 7,32, 9)( 4, 8,31,10)(13,27,18,23)(14,28,17,24) (15,25,20,21)(16,26,19,22)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1, 7, 3, 5)( 2, 8, 4, 6)( 9,32,11,30)(10,31,12,29)(13,26,15,28)(14,25,16,27) (17,21,19,23)(18,22,20,24)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1, 7, 4, 6)( 2, 8, 3, 5)( 9,31,12,30)(10,32,11,29)(13,23,16,22)(14,24,15,21) (17,28,20,25)(18,27,19,26)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1, 7,29,10)( 2, 8,30, 9)( 3, 5,31,12)( 4, 6,32,11)(13,24,17,27)(14,23,18,28) (15,22,19,25)(16,21,20,26)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1, 7,30, 9)( 2, 8,29,10)( 3, 5,32,11)( 4, 6,31,12)(13,25,18,21)(14,26,17,22) (15,27,20,23)(16,28,19,24)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,13,30,18)( 2,14,29,17)( 3,15,32,20)( 4,16,31,19)( 5,21,11,25)( 6,22,12,26) ( 7,23, 9,27)( 8,24,10,28)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,13, 4,16)( 2,14, 3,15)( 5,22, 8,23)( 6,21, 7,24)( 9,28,12,25)(10,27,11,26) (17,32,20,29)(18,31,19,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,13,29,17)( 2,14,30,18)( 3,15,31,19)( 4,16,32,20)( 5,27,12,24)( 6,28,11,23) ( 7,25,10,22)( 8,26, 9,21)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,13, 3,15)( 2,14, 4,16)( 5,28, 7,26)( 6,27, 8,25)( 9,22,11,24)(10,21,12,23) (17,31,19,29)(18,32,20,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,15,30,20)( 2,16,29,19)( 3,13,32,18)( 4,14,31,17)( 5,23,11,27)( 6,24,12,28) ( 7,21, 9,25)( 8,22,10,26)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,15, 4,14)( 2,16, 3,13)( 5,24, 8,21)( 6,23, 7,22)( 9,26,12,27)(10,25,11,28) (17,30,20,31)(18,29,19,32)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,15,29,19)( 2,16,30,20)( 3,13,31,17)( 4,14,32,18)( 5,25,12,22)( 6,26,11,21) ( 7,27,10,24)( 8,28, 9,23)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,15, 3,13)( 2,16, 4,14)( 5,26, 7,28)( 6,25, 8,27)( 9,24,11,22)(10,23,12,21) (17,29,19,31)(18,30,20,32)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,21, 2,22)( 3,23, 4,24)( 5,15, 6,16)( 7,13, 8,14)( 9,18,10,17)(11,20,12,19) (25,29,26,30)(27,31,28,32)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,21,31,28)( 2,22,32,27)( 3,23,29,26)( 4,24,30,25)( 5,16,10,18)( 6,15, 9,17) ( 7,14,12,20)( 8,13,11,19)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1,21)( 2,22)( 3,23)( 4,24)( 5,17)( 6,18)( 7,19)( 8,20)( 9,16)(10,15)(11,14) (12,13)(25,30)(26,29)(27,32)(28,31)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,21,32,27)( 2,22,31,28)( 3,23,30,25)( 4,24,29,26)( 5,18, 9,15)( 6,17,10,16) ( 7,20,11,13)( 8,19,12,14)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,23, 2,24)( 3,21, 4,22)( 5,13, 6,14)( 7,15, 8,16)( 9,20,10,19)(11,18,12,17) (25,31,26,32)(27,29,28,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,23,31,26)( 2,24,32,25)( 3,21,29,28)( 4,22,30,27)( 5,14,10,20)( 6,13, 9,19) ( 7,16,12,18)( 8,15,11,17)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1,23)( 2,24)( 3,21)( 4,22)( 5,19)( 6,20)( 7,17)( 8,18)( 9,14)(10,13)(11,16) (12,15)(25,32)(26,31)(27,30)(28,29)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,23,32,25)( 2,24,31,26)( 3,21,30,27)( 4,22,29,28)( 5,20, 9,13)( 6,19,10,14) ( 7,18,11,15)( 8,17,12,16)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,29)( 2,30)( 3,31)( 4,32)( 5,12)( 6,11)( 7,10)( 8, 9)(13,17)(14,18)(15,19) (16,20)(21,26)(22,25)(23,28)(24,27)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,30)( 2,29)( 3,32)( 4,31)( 5,11)( 6,12)( 7, 9)( 8,10)(13,18)(14,17)(15,20) (16,19)(21,25)(22,26)(23,27)(24,28)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,31)( 2,32)( 3,29)( 4,30)( 5,10)( 6, 9)( 7,12)( 8,11)(13,19)(14,20)(15,17) (16,18)(21,28)(22,27)(23,26)(24,25)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,32)( 2,31)( 3,30)( 4,29)( 5, 9)( 6,10)( 7,11)( 8,12)(13,20)(14,19)(15,18) (16,17)(21,27)(22,28)(23,25)(24,26)$

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:	$2$
Label:	128.1107	magma: IdentifyGroup(G);
Character table:	38 x 38 character table

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	32T1006
Degree	$32$
Order	$128$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$C_2^3.C_2^4$