LMFDB - Galois group: 32T44

Show commands: Magma

magma:G := TransitiveGroup(32, 44);

Group invariants

Abstract group:	$C_4:C_8$	magma:IdentifyGroup(G);
Order:	$32=2^{5}$	magma:Order(G);
Cyclic:	no	magma:IsCyclic(G);
Abelian:	no	magma:IsAbelian(G);
Solvable:	yes	magma:IsSolvable(G);
Nilpotency class:	$2$	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$32$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$44$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$32$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,5,9,13,17,23,25,29)(2,6,10,14,18,24,26,30)(3,8,11,16,19,22,27,32)(4,7,12,15,20,21,28,31)$, $(1,22,9,32,17,8,25,16)(2,21,10,31,18,7,26,15)(3,24,11,30,19,6,27,14)(4,23,12,29,20,5,28,13)$	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, $C_2^2$
$8$: $D_{4}$, $C_8$ x 2, $C_4\times C_2$, $Q_8$
$16$: $C_8:C_2$, $C_8\times C_2$, $C_4:C_4$

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, $C_2^2$
$8$:	$D_{4}$, $C_8$ x 2, $C_4\times C_2$, $Q_8$
$16$:	$C_8:C_2$, $C_8\times C_2$, $C_4:C_4$

Subfields

Degree 2: $C_2$ x 3
Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 2
Degree 8: $C_8$ x 2, $C_4\times C_2$, $D_4$, $Q_8$, $C_8:C_2$
Degree 16: $C_8\times C_2$, $C_8: C_2$, $C_4:C_4$

Low degree siblings

There are no siblings 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	$2^{16}$	$1$	$2$	$16$	$( 1,17)( 2,18)( 3,19)( 4,20)( 5,23)( 6,24)( 7,21)( 8,22)( 9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)$
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	$2^{16}$	$1$	$2$	$16$	$( 1,18)( 2,17)( 3,20)( 4,19)( 5,24)( 6,23)( 7,22)( 8,21)( 9,26)(10,25)(11,28)(12,27)(13,30)(14,29)(15,32)(16,31)$
4A1	$4^{8}$	$1$	$4$	$24$	$( 1,10,17,26)( 2, 9,18,25)( 3,12,19,28)( 4,11,20,27)( 5,14,23,30)( 6,13,24,29)( 7,16,21,32)( 8,15,22,31)$
4A-1	$4^{8}$	$1$	$4$	$24$	$( 1,26,17,10)( 2,25,18, 9)( 3,28,19,12)( 4,27,20,11)( 5,30,23,14)( 6,29,24,13)( 7,32,21,16)( 8,31,22,15)$
4B1	$4^{8}$	$1$	$4$	$24$	$( 1, 9,17,25)( 2,10,18,26)( 3,11,19,27)( 4,12,20,28)( 5,13,23,29)( 6,14,24,30)( 7,15,21,31)( 8,16,22,32)$
4B-1	$4^{8}$	$1$	$4$	$24$	$( 1,25,17, 9)( 2,26,18,10)( 3,27,19,11)( 4,28,20,12)( 5,29,23,13)( 6,30,24,14)( 7,31,21,15)( 8,32,22,16)$
4C	$4^{8}$	$2$	$4$	$24$	$( 1,19, 2,20)( 3,18, 4,17)( 5,21, 6,22)( 7,24, 8,23)( 9,27,10,28)(11,26,12,25)(13,31,14,32)(15,30,16,29)$
4D	$4^{8}$	$2$	$4$	$24$	$( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)(17,19,18,20)(21,24,22,23)(25,27,26,28)(29,31,30,32)$
4E1	$4^{8}$	$2$	$4$	$24$	$( 1,27,18,12)( 2,28,17,11)( 3,26,20, 9)( 4,25,19,10)( 5,31,24,16)( 6,32,23,15)( 7,30,22,13)( 8,29,21,14)$
4E-1	$4^{8}$	$2$	$4$	$24$	$( 1,11,18,28)( 2,12,17,27)( 3,10,20,25)( 4, 9,19,26)( 5,15,24,32)( 6,16,23,31)( 7,14,22,29)( 8,13,21,30)$
8A1	$8^{4}$	$2$	$8$	$28$	$( 1, 5, 9,13,17,23,25,29)( 2, 6,10,14,18,24,26,30)( 3, 8,11,16,19,22,27,32)( 4, 7,12,15,20,21,28,31)$
8A-1	$8^{4}$	$2$	$8$	$28$	$( 1,29,25,23,17,13, 9, 5)( 2,30,26,24,18,14,10, 6)( 3,32,27,22,19,16,11, 8)( 4,31,28,21,20,15,12, 7)$
8A3	$8^{4}$	$2$	$8$	$28$	$( 1,13,25, 5,17,29, 9,23)( 2,14,26, 6,18,30,10,24)( 3,16,27, 8,19,32,11,22)( 4,15,28, 7,20,31,12,21)$
8A-3	$8^{4}$	$2$	$8$	$28$	$( 1,23, 9,29,17, 5,25,13)( 2,24,10,30,18, 6,26,14)( 3,22,11,32,19, 8,27,16)( 4,21,12,31,20, 7,28,15)$
8B1	$8^{4}$	$2$	$8$	$28$	$( 1,21, 9,31,17, 7,25,15)( 2,22,10,32,18, 8,26,16)( 3,23,11,29,19, 5,27,13)( 4,24,12,30,20, 6,28,14)$
8B-1	$8^{4}$	$2$	$8$	$28$	$( 1,15,25, 7,17,31, 9,21)( 2,16,26, 8,18,32,10,22)( 3,13,27, 5,19,29,11,23)( 4,14,28, 6,20,30,12,24)$
8B3	$8^{4}$	$2$	$8$	$28$	$( 1,31,25,21,17,15, 9, 7)( 2,32,26,22,18,16,10, 8)( 3,29,27,23,19,13,11, 5)( 4,30,28,24,20,14,12, 6)$
8B-3	$8^{4}$	$2$	$8$	$28$	$( 1, 7, 9,15,17,21,25,31)( 2, 8,10,16,18,22,26,32)( 3, 5,11,13,19,23,27,29)( 4, 6,12,14,20,24,28,30)$

Malle's constant $a(G)$: $1/16$

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	4A1	4A-1	4B1	4B-1	4C	4D	4E1	4E-1	8A1	8A-1	8A3	8A-3	8B1	8B-1	8B3	8B-3
	Size	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2
	2 P	1A	1A	1A	1A	2A	2A	2A	2A	2B	2B	2C	2C	4B1	4B-1	4B-1	4B1	4B1	4B-1	4B-1	4B1
	Type
32.12.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.12.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$
32.12.1c	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
32.12.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
32.12.1e1	C	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- i$	$i$	$i$	$- i$	$i$	$- i$	$- i$	$i$
32.12.1e2	C	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$i$	$- i$	$- i$	$i$	$- i$	$i$	$i$	$- i$
32.12.1f1	C	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- i$	$i$	$i$	$- i$	$- i$	$i$	$i$	$- i$
32.12.1f2	C	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$i$	$- i$	$- i$	$i$	$i$	$- i$	$- i$	$i$
32.12.1g1	C	$1$	$- 1$	$1$	$- 1$	$- ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}^{2}$	$- 1$	$1$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}$	$ζ_{8}^{3}$	$- ζ_{8}^{3}$	$ζ_{8}$	$- ζ_{8}$	$ζ_{8}^{3}$	$- ζ_{8}^{3}$	$ζ_{8}$
32.12.1g2	C	$1$	$- 1$	$1$	$- 1$	$ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- 1$	$1$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}^{3}$	$- ζ_{8}$	$ζ_{8}$	$- ζ_{8}^{3}$	$ζ_{8}^{3}$	$- ζ_{8}$	$ζ_{8}$	$- ζ_{8}^{3}$
32.12.1g3	C	$1$	$- 1$	$1$	$- 1$	$- ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}^{2}$	$- 1$	$1$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}$	$- ζ_{8}^{3}$	$ζ_{8}^{3}$	$- ζ_{8}$	$ζ_{8}$	$- ζ_{8}^{3}$	$ζ_{8}^{3}$	$- ζ_{8}$
32.12.1g4	C	$1$	$- 1$	$1$	$- 1$	$ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- 1$	$1$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{3}$	$ζ_{8}$	$- ζ_{8}$	$ζ_{8}^{3}$	$- ζ_{8}^{3}$	$ζ_{8}$	$- ζ_{8}$	$ζ_{8}^{3}$
32.12.1h1	C	$1$	$- 1$	$1$	$- 1$	$- ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}^{2}$	$1$	$- 1$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}$	$ζ_{8}^{3}$	$- ζ_{8}^{3}$	$ζ_{8}$	$ζ_{8}$	$- ζ_{8}^{3}$	$ζ_{8}^{3}$	$- ζ_{8}$
32.12.1h2	C	$1$	$- 1$	$1$	$- 1$	$ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}^{2}$	$1$	$- 1$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{3}$	$- ζ_{8}$	$ζ_{8}$	$- ζ_{8}^{3}$	$- ζ_{8}^{3}$	$ζ_{8}$	$- ζ_{8}$	$ζ_{8}^{3}$
32.12.1h3	C	$1$	$- 1$	$1$	$- 1$	$- ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}^{2}$	$1$	$- 1$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}$	$- ζ_{8}^{3}$	$ζ_{8}^{3}$	$- ζ_{8}$	$- ζ_{8}$	$ζ_{8}^{3}$	$- ζ_{8}^{3}$	$ζ_{8}$
32.12.1h4	C	$1$	$- 1$	$1$	$- 1$	$ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}^{2}$	$1$	$- 1$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}^{3}$	$ζ_{8}$	$- ζ_{8}$	$ζ_{8}^{3}$	$ζ_{8}^{3}$	$- ζ_{8}$	$ζ_{8}$	$- ζ_{8}^{3}$
32.12.2a	R	$2$	$2$	$- 2$	$- 2$	$2$	$- 2$	$- 2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.12.2b	S	$2$	$2$	$- 2$	$- 2$	$- 2$	$2$	$2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.12.2c1	C	$2$	$- 2$	$- 2$	$2$	$- 2 i$	$2 i$	$- 2 i$	$2 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.12.2c2	C	$2$	$- 2$	$- 2$	$2$	$2 i$	$- 2 i$	$2 i$	$- 2 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

magma:CharacterTable(G);

Regular extensions

Data not computed

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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	32T44

Degree	$32$
Order	$32$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$C_4:C_8$