LMFDB - Galois group: 32T273

Show commands: Magma

magma: G := TransitiveGroup(32, 273);

Group action invariants

Degree $n$:	$32$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$273$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$D_4\times C_2^3$
Parity:	$1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$16$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,20)(14,19)(15,18)(16,17)(21,28)(22,27)(23,26)(24,25)(29,30)(31,32), (1,20,29,14)(2,19,30,13)(3,18,31,16)(4,17,32,15)(5,22,11,28)(6,21,12,27)(7,24,9,26)(8,23,10,25), (1,12)(2,11)(3,10)(4,9)(5,30)(6,29)(7,32)(8,31)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21), (1,24)(2,23)(3,22)(4,21)(5,18)(6,17)(7,20)(8,19)(9,14)(10,13)(11,16)(12,15)(25,30)(26,29)(27,32)(28,31), (1,13,29,19)(2,14,30,20)(3,15,31,17)(4,16,32,18)(5,27,11,21)(6,28,12,22)(7,25,9,23)(8,26,10,24)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 31
$4$: $C_2^2$ x 155
$8$: $D_{4}$ x 8, $C_2^3$ x 155
$16$: $D_4\times C_2$ x 28, $C_2^4$ x 31
$32$: $C_2^2 \times D_4$ x 14, 32T39

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 31
$4$:	$C_2^2$ x 155
$8$:	$D_{4}$ x 8, $C_2^3$ x 155
$16$:	$D_4\times C_2$ x 28, $C_2^4$ x 31
$32$:	$C_2^2 \times D_4$ x 14, 32T39

Subfields

Degree 2: $C_2$ x 15
Degree 4: $C_2^2$ x 35, $D_{4}$ x 8
Degree 8: $C_2^3$ x 15, $D_4\times C_2$ x 28
Degree 16: $C_2^4$, $C_2^2 \times D_4$ x 14

Low degree siblings

32T273 x 15
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 $	$2$	$2$	$(13,19)(14,20)(15,17)(16,18)(21,27)(22,28)(23,25)(24,26)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 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 $	$2$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,20)(14,19)(15,18)(16,17)(21,28) (22,27)(23,26)(24,25)(29,30)(31,32)$
	$ 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 $	$2$	$2$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,17)(14,18)(15,19)(16,20)(21,25) (22,26)(23,27)(24,28)(29,31)(30,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,18)(14,17)(15,20)(16,19)(21,26) (22,25)(23,28)(24,27)(29,32)(30,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,31)(10,32)(11,29)(12,30)(13,21)(14,22)(15,23) (16,24)(17,25)(18,26)(19,27)(20,28)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,31)(10,32)(11,29)(12,30)(13,27)(14,28)(15,25) (16,26)(17,23)(18,24)(19,21)(20,22)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,32)(10,31)(11,30)(12,29)(13,22)(14,21)(15,24) (16,23)(17,26)(18,25)(19,28)(20,27)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26) (16,25)(17,24)(18,23)(19,22)(20,21)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,29)(10,30)(11,31)(12,32)(13,23)(14,24)(15,21) (16,22)(17,27)(18,28)(19,25)(20,26)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27) (16,28)(17,21)(18,22)(19,23)(20,24)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,30)(10,29)(11,32)(12,31)(13,24)(14,23)(15,22) (16,21)(17,28)(18,27)(19,26)(20,25)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,30)(10,29)(11,32)(12,31)(13,26)(14,25)(15,28) (16,27)(17,22)(18,21)(19,24)(20,23)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 9)( 2,10)( 3,11)( 4,12)( 5,31)( 6,32)( 7,29)( 8,30)(13,23)(14,24)(15,21) (16,22)(17,27)(18,28)(19,25)(20,26)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,10)( 2, 9)( 3,12)( 4,11)( 5,32)( 6,31)( 7,30)( 8,29)(13,24)(14,23)(15,22) (16,21)(17,28)(18,27)(19,26)(20,25)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,11)( 2,12)( 3, 9)( 4,10)( 5,29)( 6,30)( 7,31)( 8,32)(13,21)(14,22)(15,23) (16,24)(17,25)(18,26)(19,27)(20,28)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,12)( 2,11)( 3,10)( 4, 9)( 5,30)( 6,29)( 7,32)( 8,31)(13,22)(14,21)(15,24) (16,23)(17,26)(18,25)(19,28)(20,27)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1,13)( 2,14)( 3,15)( 4,16)( 5,27)( 6,28)( 7,25)( 8,26)( 9,23)(10,24)(11,21) (12,22)(17,31)(18,32)(19,29)(20,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,13,29,19)( 2,14,30,20)( 3,15,31,17)( 4,16,32,18)( 5,27,11,21)( 6,28,12,22) ( 7,25, 9,23)( 8,26,10,24)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1,14)( 2,13)( 3,16)( 4,15)( 5,28)( 6,27)( 7,26)( 8,25)( 9,24)(10,23)(11,22) (12,21)(17,32)(18,31)(19,30)(20,29)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,14,29,20)( 2,13,30,19)( 3,16,31,18)( 4,15,32,17)( 5,28,11,22)( 6,27,12,21) ( 7,26, 9,24)( 8,25,10,23)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1,15)( 2,16)( 3,13)( 4,14)( 5,25)( 6,26)( 7,27)( 8,28)( 9,21)(10,22)(11,23) (12,24)(17,29)(18,30)(19,31)(20,32)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,15,29,17)( 2,16,30,18)( 3,13,31,19)( 4,14,32,20)( 5,25,11,23)( 6,26,12,24) ( 7,27, 9,21)( 8,28,10,22)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1,16)( 2,15)( 3,14)( 4,13)( 5,26)( 6,25)( 7,28)( 8,27)( 9,22)(10,21)(11,24) (12,23)(17,30)(18,29)(19,32)(20,31)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,16,29,18)( 2,15,30,17)( 3,14,31,20)( 4,13,32,19)( 5,26,11,24)( 6,25,12,23) ( 7,28, 9,22)( 8,27,10,21)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,21,29,27)( 2,22,30,28)( 3,23,31,25)( 4,24,32,26)( 5,19,11,13)( 6,20,12,14) ( 7,17, 9,15)( 8,18,10,16)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1,21)( 2,22)( 3,23)( 4,24)( 5,19)( 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,13) (12,14)(25,31)(26,32)(27,29)(28,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,22,29,28)( 2,21,30,27)( 3,24,31,26)( 4,23,32,25)( 5,20,11,14)( 6,19,12,13) ( 7,18, 9,16)( 8,17,10,15)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1,22)( 2,21)( 3,24)( 4,23)( 5,20)( 6,19)( 7,18)( 8,17)( 9,16)(10,15)(11,14) (12,13)(25,32)(26,31)(27,30)(28,29)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,23,29,25)( 2,24,30,26)( 3,21,31,27)( 4,22,32,28)( 5,17,11,15)( 6,18,12,16) ( 7,19, 9,13)( 8,20,10,14)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1,23)( 2,24)( 3,21)( 4,22)( 5,17)( 6,18)( 7,19)( 8,20)( 9,13)(10,14)(11,15) (12,16)(25,29)(26,30)(27,31)(28,32)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,24,29,26)( 2,23,30,25)( 3,22,31,28)( 4,21,32,27)( 5,18,11,16)( 6,17,12,15) ( 7,20, 9,14)( 8,19,10,13)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1,24)( 2,23)( 3,22)( 4,21)( 5,18)( 6,17)( 7,20)( 8,19)( 9,14)(10,13)(11,16) (12,15)(25,30)(26,29)(27,32)(28,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,29)( 2,30)( 3,31)( 4,32)( 5,11)( 6,12)( 7, 9)( 8,10)(13,19)(14,20)(15,17) (16,18)(21,27)(22,28)(23,25)(24,26)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,30)( 2,29)( 3,32)( 4,31)( 5,12)( 6,11)( 7,10)( 8, 9)(13,20)(14,19)(15,18) (16,17)(21,28)(22,27)(23,26)(24,25)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,31)( 2,32)( 3,29)( 4,30)( 5, 9)( 6,10)( 7,11)( 8,12)(13,17)(14,18)(15,19) (16,20)(21,25)(22,26)(23,27)(24,28)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,32)( 2,31)( 3,30)( 4,29)( 5,10)( 6, 9)( 7,12)( 8,11)(13,18)(14,17)(15,20) (16,19)(21,26)(22,25)(23,28)(24,27)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$64=2^{6}$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	$2$
Label:	64.261	magma: IdentifyGroup(G);
Character table:	40 x 40 character table

magma: CharacterTable(G);

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Group action invariants

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Low degree siblings

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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	32T273
Degree	$32$
Order	$64$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$D_4\times C_2^3$