LMFDB - Galois group: 32T339

Show commands: Magma

magma:G := TransitiveGroup(32, 339);

Group invariants

Abstract group:	$C_8:D_4$	magma:IdentifyGroup(G);
Order:	$64=2^{6}$	magma:Order(G);
Cyclic:	no	magma:IsCyclic(G);
Abelian:	no	magma:IsAbelian(G);
Solvable:	yes	magma:IsSolvable(G);
Nilpotency class:	$3$	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$32$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$339$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$4$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(3,4)(5,13)(6,14)(7,15)(8,16)(9,27)(10,28)(11,25)(12,26)(17,18)(21,29)(22,30)(23,32)(24,31)$, $(1,5)(2,6)(3,7)(4,8)(9,29)(10,30)(11,32)(12,31)(13,26)(14,25)(15,27)(16,28)(17,21)(18,22)(19,23)(20,24)$, $(1,16,25,8,20,29,11,21)(2,15,26,7,19,30,12,22)(3,13,27,6,18,31,10,23)(4,14,28,5,17,32,9,24)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 7
$4$: $C_2^2$ x 7
$8$: $D_{4}$ x 6, $C_2^3$
$16$: $D_4\times C_2$ x 3
$32$: $Z_8 : Z_8^\times$ x 2, $C_4^2:C_2$

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_2^2$ x 7
$8$:	$D_{4}$ x 6, $C_2^3$
$16$:	$D_4\times C_2$ x 3
$32$:	$Z_8 : Z_8^\times$ x 2, $C_4^2:C_2$

Subfields

Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 6
Degree 8: $D_4\times C_2$ x 3, $Z_8 : Z_8^\times$ x 4
Degree 16: 16T38 x 2, $C_4^2:C_2$

Low degree siblings

32T339, 32T360
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	Index	Representative
1A	$1^{32}$	$1$	$1$	$0$	$()$
2A	$2^{16}$	$1$	$2$	$16$	$( 1,19)( 2,20)( 3,17)( 4,18)( 5,23)( 6,24)( 7,21)( 8,22)( 9,27)(10,28)(11,26)(12,25)(13,32)(14,31)(15,29)(16,30)$
2B	$2^{16}$	$1$	$2$	$16$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2C	$2^{16}$	$1$	$2$	$16$	$( 1,20)( 2,19)( 3,18)( 4,17)( 5,24)( 6,23)( 7,22)( 8,21)( 9,28)(10,27)(11,25)(12,26)(13,31)(14,32)(15,30)(16,29)$
2D	$2^{14},1^{4}$	$8$	$2$	$14$	$( 1,12)( 2,11)( 3,10)( 4, 9)( 5,24)( 6,23)( 7,21)( 8,22)(15,16)(17,28)(18,27)(19,25)(20,26)(29,30)$
2E	$2^{16}$	$8$	$2$	$16$	$( 1, 3)( 2, 4)( 5,30)( 6,29)( 7,32)( 8,31)( 9,26)(10,25)(11,27)(12,28)(13,21)(14,22)(15,24)(16,23)(17,19)(18,20)$
2F	$2^{16}$	$8$	$2$	$16$	$( 1,24)( 2,23)( 3,22)( 4,21)( 5,20)( 6,19)( 7,18)( 8,17)( 9,16)(10,15)(11,14)(12,13)(25,32)(26,31)(27,30)(28,29)$
4A	$4^{8}$	$2$	$4$	$24$	$( 1,11,20,25)( 2,12,19,26)( 3,10,18,27)( 4, 9,17,28)( 5,14,24,32)( 6,13,23,31)( 7,15,22,30)( 8,16,21,29)$
4B	$4^{8}$	$2$	$4$	$24$	$( 1,12,20,26)( 2,11,19,25)( 3, 9,18,28)( 4,10,17,27)( 5,13,24,31)( 6,14,23,32)( 7,16,22,29)( 8,15,21,30)$
4C	$4^{8}$	$4$	$4$	$24$	$( 1,28,19,10)( 2,27,20, 9)( 3,25,17,12)( 4,26,18,11)( 5,15,23,29)( 6,16,24,30)( 7,13,21,32)( 8,14,22,31)$
4D	$4^{8}$	$4$	$4$	$24$	$( 1, 4, 2, 3)( 5,22, 6,21)( 7,23, 8,24)( 9,12,10,11)(13,29,14,30)(15,31,16,32)(17,19,18,20)(25,28,26,27)$
4E	$4^{8}$	$8$	$4$	$24$	$( 1,29,20,16)( 2,30,19,15)( 3,32,18,14)( 4,31,17,13)( 5,10,24,27)( 6, 9,23,28)( 7,12,22,26)( 8,11,21,25)$
8A	$8^{4}$	$4$	$8$	$28$	$( 1,22,11,30,20, 7,25,15)( 2,21,12,29,19, 8,26,16)( 3,24,10,32,18, 5,27,14)( 4,23, 9,31,17, 6,28,13)$
8B	$8^{4}$	$4$	$8$	$28$	$( 1, 8,11,16,20,21,25,29)( 2, 7,12,15,19,22,26,30)( 3, 6,10,13,18,23,27,31)( 4, 5, 9,14,17,24,28,32)$
8C	$8^{4}$	$4$	$8$	$28$	$( 1,13,12,24,20,31,26, 5)( 2,14,11,23,19,32,25, 6)( 3,15, 9,21,18,30,28, 8)( 4,16,10,22,17,29,27, 7)$
8D	$8^{4}$	$4$	$8$	$28$	$( 1,23,26,14,20, 6,12,32)( 2,24,25,13,19, 5,11,31)( 3,22,28,16,18, 7, 9,29)( 4,21,27,15,17, 8,10,30)$

Malle's constant $a(G)$: $1/14$

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	8A	8B	8C	8D
	Size	1	1	1	1	8	8	8	2	2	4	4	8	4	4	4	4
	2 P	1A	1A	1A	1A	1A	1A	1A	2C	2C	2A	2B	2C	4A	4A	4B	4B
	Type
64.177.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
64.177.1b	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$
64.177.1c	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
64.177.1d	R	$1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$
64.177.1e	R	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$
64.177.1f	R	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$
64.177.1g	R	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$
64.177.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
64.177.2a	R	$2$	$2$	$- 2$	$- 2$	$0$	$0$	$0$	$- 2$	$2$	$0$	$0$	$0$	$0$	$- 2$	$2$	$0$
64.177.2b	R	$2$	$2$	$- 2$	$- 2$	$0$	$0$	$0$	$- 2$	$2$	$0$	$0$	$0$	$0$	$2$	$- 2$	$0$
64.177.2c	R	$2$	$2$	$- 2$	$- 2$	$0$	$0$	$0$	$2$	$- 2$	$0$	$0$	$0$	$- 2$	$0$	$0$	$2$
64.177.2d	R	$2$	$2$	$- 2$	$- 2$	$0$	$0$	$0$	$2$	$- 2$	$0$	$0$	$0$	$2$	$0$	$0$	$- 2$
64.177.2e	R	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$- 2$	$- 2$	$- 2$	$2$	$0$	$0$	$0$	$0$	$0$
64.177.2f	R	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$- 2$	$- 2$	$2$	$- 2$	$0$	$0$	$0$	$0$	$0$
64.177.4a	R	$4$	$- 4$	$- 4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.177.4b	R	$4$	$- 4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

magma:CharacterTable(G);

Regular extensions

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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	32T339

Degree	$32$
Order	$64$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$C_8:D_4$