LMFDB - Galois group: 32T108

Show commands: Magma

magma:G := TransitiveGroup(32, 108);

Group invariants

Abstract group:	$\OD_{32}:C_2$	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 $t$:	$108$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$8$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,7,3,6)(2,8,4,5)(9,16,11,13)(10,15,12,14)(17,22,20,24)(18,21,19,23)(25,29,27,31)(26,30,28,32)$, $(1,15,23,32,5,10,17,26,3,14,21,30,8,12,20,28)(2,16,24,31,6,9,18,25,4,13,22,29,7,11,19,27)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_4$ x 2, $C_2^2$
$8$: $D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$
$16$: $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$
$32$: $C_2^2 : C_8$

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$8$:	$D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$
$16$:	$C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$
$32$:	$C_2^2 : C_8$

Subfields

Degree 2: $C_2$
Degree 4: $C_4$, $D_{4}$ x 2
Degree 8: $C_8$, $C_8:C_2$, $C_2^2:C_4$
Degree 16: $C_2^2 : C_8$, 16T104

Low degree siblings

16T104, 32T107
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	$2^{16}$	$1$	$2$	$16$	$( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$
2B	$2^{16}$	$2$	$2$	$16$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)$
2C	$2^{12},1^{8}$	$4$	$2$	$12$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28)(29,31)(30,32)$
4A1	$4^{8}$	$1$	$4$	$24$	$( 1, 8, 3, 5)( 2, 7, 4, 6)( 9,16,11,13)(10,15,12,14)(17,23,20,21)(18,24,19,22)(25,31,27,29)(26,32,28,30)$
4A-1	$4^{8}$	$1$	$4$	$24$	$( 1, 5, 3, 8)( 2, 6, 4, 7)( 9,13,11,16)(10,14,12,15)(17,21,20,23)(18,22,19,24)(25,29,27,31)(26,30,28,32)$
4B	$4^{8}$	$2$	$4$	$24$	$( 1, 7, 3, 6)( 2, 8, 4, 5)( 9,14,11,15)(10,13,12,16)(17,24,20,22)(18,23,19,21)(25,30,27,32)(26,29,28,31)$
4C	$4^{8}$	$4$	$4$	$24$	$( 1, 6, 3, 7)( 2, 5, 4, 8)( 9,13,11,16)(10,14,12,15)(17,24,20,22)(18,23,19,21)(25,31,27,29)(26,32,28,30)$
8A1	$8^{4}$	$2$	$8$	$28$	$( 1,23, 5,17, 3,21, 8,20)( 2,24, 6,18, 4,22, 7,19)( 9,25,13,29,11,27,16,31)(10,26,14,30,12,28,15,32)$
8A-1	$8^{4}$	$2$	$8$	$28$	$( 1,20, 8,21, 3,17, 5,23)( 2,19, 7,22, 4,18, 6,24)( 9,31,16,27,11,29,13,25)(10,32,15,28,12,30,14,26)$
8B1	$8^{4}$	$2$	$8$	$28$	$( 1,18, 8,24, 3,19, 5,22)( 2,17, 7,23, 4,20, 6,21)( 9,32,16,28,11,30,13,26)(10,31,15,27,12,29,14,25)$
8B-1	$8^{4}$	$2$	$8$	$28$	$( 1,22, 5,19, 3,24, 8,18)( 2,21, 6,20, 4,23, 7,17)( 9,26,13,30,11,28,16,32)(10,25,14,29,12,27,15,31)$
8C1	$8^{4}$	$4$	$8$	$28$	$( 1,22, 8,18, 3,24, 5,19)( 2,21, 7,17, 4,23, 6,20)( 9,27,16,29,11,25,13,31)(10,28,15,30,12,26,14,32)$
8C-1	$8^{4}$	$4$	$8$	$28$	$( 1,18, 5,22, 3,19, 8,24)( 2,17, 6,21, 4,20, 7,23)( 9,29,13,27,11,31,16,25)(10,30,14,28,12,32,15,26)$
16A1	$16^{2}$	$4$	$16$	$30$	$( 1,15,23,32, 5,10,17,26, 3,14,21,30, 8,12,20,28)( 2,16,24,31, 6, 9,18,25, 4,13,22,29, 7,11,19,27)$
16A-1	$16^{2}$	$4$	$16$	$30$	$( 1,28,20,12, 8,30,21,14, 3,26,17,10, 5,32,23,15)( 2,27,19,11, 7,29,22,13, 4,25,18, 9, 6,31,24,16)$
16A3	$16^{2}$	$4$	$16$	$30$	$( 1,32,17,14, 8,28,23,10, 3,30,20,15, 5,26,21,12)( 2,31,18,13, 7,27,24, 9, 4,29,19,16, 6,25,22,11)$
16A-3	$16^{2}$	$4$	$16$	$30$	$( 1,12,21,26, 5,15,20,30, 3,10,23,28, 8,14,17,32)( 2,11,22,25, 6,16,19,29, 4, 9,24,27, 7,13,18,31)$
16B1	$16^{2}$	$4$	$16$	$30$	$( 1,30,18,13, 8,26,24, 9, 3,32,19,16, 5,28,22,11)( 2,29,17,14, 7,25,23,10, 4,31,20,15, 6,27,21,12)$
16B-1	$16^{2}$	$4$	$16$	$30$	$( 1,10,24,27, 5,14,18,31, 3,12,22,25, 8,15,19,29)( 2, 9,23,28, 6,13,17,32, 4,11,21,26, 7,16,20,30)$
16B3	$16^{2}$	$4$	$16$	$30$	$( 1,15,22,31, 5,10,19,25, 3,14,24,29, 8,12,18,27)( 2,16,21,32, 6, 9,20,26, 4,13,23,30, 7,11,17,28)$
16B-3	$16^{2}$	$4$	$16$	$30$	$( 1,26,19,11, 8,32,22,13, 3,28,18, 9, 5,30,24,16)( 2,25,20,12, 7,31,21,14, 4,27,17,10, 6,29,23,15)$

Malle's constant $a(G)$: $1/12$

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	4A1	4A-1	4B	4C	8A1	8A-1	8B1	8B-1	8C1	8C-1	16A1	16A-1	16A3	16A-3	16B1	16B-1	16B3	16B-3
	Size	1	1	2	4	1	1	2	4	2	2	2	2	4	4	4	4	4	4	4	4	4	4
	2 P	1A	1A	1A	1A	2A	2A	2A	2A	4A-1	4A1	4A1	4A-1	4A1	4A-1	8A1	8A-1	8A-1	8A1	8B1	8B-1	8B-1	8B1
	Type
64.30.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
64.30.1b	R	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$
64.30.1c	R	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$
64.30.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
64.30.1e1	C	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- i$	$- i$	$i$	$i$	$- i$	$i$	$- i$	$i$
64.30.1e2	C	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$i$	$i$	$- i$	$- i$	$i$	$- i$	$i$	$- i$
64.30.1f1	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- i$	$i$	$- i$	$i$	$i$	$- i$	$- i$	$i$
64.30.1f2	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$i$	$- i$	$i$	$- i$	$- i$	$i$	$i$	$- i$
64.30.1g1	C	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}$	$- ζ_{8}^{3}$	$- ζ_{8}$	$- ζ_{8}^{3}$	$ζ_{8}^{3}$	$ζ_{8}$	$ζ_{8}$	$ζ_{8}^{3}$
64.30.1g2	C	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}^{3}$	$ζ_{8}$	$ζ_{8}^{3}$	$ζ_{8}$	$- ζ_{8}$	$- ζ_{8}^{3}$	$- ζ_{8}^{3}$	$- ζ_{8}$
64.30.1g3	C	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}$	$ζ_{8}^{3}$	$ζ_{8}$	$ζ_{8}^{3}$	$- ζ_{8}^{3}$	$- ζ_{8}$	$- ζ_{8}$	$- ζ_{8}^{3}$
64.30.1g4	C	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{3}$	$- ζ_{8}$	$- ζ_{8}^{3}$	$- ζ_{8}$	$ζ_{8}$	$ζ_{8}^{3}$	$ζ_{8}^{3}$	$ζ_{8}$
64.30.1h1	C	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}$	$ζ_{8}^{3}$	$ζ_{8}$	$- ζ_{8}^{3}$	$- ζ_{8}^{3}$	$- ζ_{8}$	$ζ_{8}$	$ζ_{8}^{3}$
64.30.1h2	C	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{3}$	$- ζ_{8}$	$- ζ_{8}^{3}$	$ζ_{8}$	$ζ_{8}$	$ζ_{8}^{3}$	$- ζ_{8}^{3}$	$- ζ_{8}$
64.30.1h3	C	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}$	$- ζ_{8}^{3}$	$- ζ_{8}$	$ζ_{8}^{3}$	$ζ_{8}^{3}$	$ζ_{8}$	$- ζ_{8}$	$- ζ_{8}^{3}$
64.30.1h4	C	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}^{3}$	$ζ_{8}$	$ζ_{8}^{3}$	$- ζ_{8}$	$- ζ_{8}$	$- ζ_{8}^{3}$	$ζ_{8}^{3}$	$ζ_{8}$
64.30.2a	R	$2$	$2$	$- 2$	$0$	$2$	$2$	$- 2$	$0$	$- 2$	$- 2$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.30.2b	R	$2$	$2$	$- 2$	$0$	$2$	$2$	$- 2$	$0$	$2$	$2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.30.2c1	C	$2$	$2$	$- 2$	$0$	$- 2$	$- 2$	$2$	$0$	$- 2 i$	$2 i$	$- 2 i$	$2 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.30.2c2	C	$2$	$2$	$- 2$	$0$	$- 2$	$- 2$	$2$	$0$	$2 i$	$- 2 i$	$2 i$	$- 2 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.30.4a1	C	$4$	$- 4$	$0$	$0$	$- 4 i$	$4 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.30.4a2	C	$4$	$- 4$	$0$	$0$	$4 i$	$- 4 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

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	32T108

Degree	$32$
Order	$64$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$\OD_{32}:C_2$