LMFDB - Galois group: 32T7297

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magma:G := TransitiveGroup(32, 7297);

Group invariants

Abstract group:	$(C_4\times C_8).D_4$	magma:IdentifyGroup(G);
Order:	$256=2^{8}$	magma:Order(G);
Cyclic:	no	magma:IsCyclic(G);
Abelian:	no	magma:IsAbelian(G);
Solvable:	yes	magma:IsSolvable(G);
Nilpotency class:	$6$	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$32$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$7297$	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:	$(1,27)(2,28)(3,26)(4,25)(5,14)(6,13)(7,23)(8,24)(9,12)(10,11)(15,29)(16,30)(17,20)(18,19)(21,32)(22,31)$, $(1,21)(2,22)(3,15,4,16)(5,18,7,11)(6,17,8,12)(9,31,10,32)(13,26,23,27)(14,25,24,28)(19,30)(20,29)$	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, $C_4\times C_2$
$16$: $C_2^2:C_4$
$32$: $C_2^3 : C_4 $
$64$: $((C_8 : C_2):C_2):C_2$
$128$: 16T335

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, $C_4\times C_2$
$16$:	$C_2^2:C_4$
$32$:	$C_2^3 : C_4 $
$64$:	$((C_8 : C_2):C_2):C_2$
$128$:	16T335

Subfields

Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 8: $D_4$, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2
Degree 16: 16T171

Low degree siblings

16T658 x 2, 32T3214, 32T3215, 32T3216, 32T7128, 32T7142, 32T7145
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	$2^{16}$	$1$	$2$	$16$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2B	$2^{8},1^{16}$	$2$	$2$	$8$	$( 1, 2)( 3, 4)( 9,10)(11,12)(17,18)(19,20)(25,26)(27,28)$
2C	$2^{16}$	$16$	$2$	$16$	$( 1, 9)( 2,10)( 3,19)( 4,20)( 5,32)( 6,31)( 7,30)( 8,29)(11,27)(12,28)(13,21)(14,22)(15,23)(16,24)(17,26)(18,25)$
2D	$2^{12},1^{8}$	$16$	$2$	$12$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,19)(10,20)(13,24)(14,23)(17,18)(21,22)(27,28)(29,30)$
4A	$4^{4},1^{16}$	$4$	$4$	$12$	$( 1, 4, 2, 3)( 9,20,10,19)(11,17,12,18)(25,27,26,28)$
4B	$4^{8}$	$4$	$4$	$24$	$( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,20,10,19)(11,17,12,18)(13,23,14,24)(15,21,16,22)(25,27,26,28)(29,32,30,31)$
4C	$4^{4},2^{8}$	$4$	$4$	$20$	$( 1, 4, 2, 3)( 5, 6)( 7, 8)( 9,20,10,19)(11,17,12,18)(13,14)(15,16)(21,22)(23,24)(25,27,26,28)(29,30)(31,32)$
4D1	$4^{4},2^{8}$	$8$	$4$	$20$	$( 1,20, 2,19)( 3, 9, 4,10)( 5,30)( 6,29)( 7,31)( 8,32)(11,26,12,25)(13,16)(14,15)(17,28,18,27)(21,23)(22,24)$
4D-1	$4^{4},2^{8}$	$8$	$4$	$20$	$( 1,19, 2,20)( 3,10, 4, 9)( 5,29)( 6,30)( 7,32)( 8,31)(11,25,12,26)(13,15)(14,16)(17,27,18,28)(21,24)(22,23)$
4E	$4^{8}$	$16$	$4$	$24$	$( 1,18, 2,17)( 3,11, 4,12)( 5,16, 6,15)( 7,22, 8,21)( 9,26,10,25)(13,30,14,29)(19,28,20,27)(23,32,24,31)$
4F1	$4^{6},2^{4}$	$32$	$4$	$22$	$( 1, 7, 4, 6)( 2, 8, 3, 5)( 9,23,19,14)(10,24,20,13)(11,16)(12,15)(17,21,18,22)(25,31)(26,32)(27,30,28,29)$
4F-1	$4^{6},2^{4}$	$32$	$4$	$22$	$( 1, 6, 4, 7)( 2, 5, 3, 8)( 9,14,19,23)(10,13,20,24)(11,16)(12,15)(17,22,18,21)(25,31)(26,32)(27,29,28,30)$
8A1	$8^{4}$	$4$	$8$	$28$	$( 1,17, 4,12, 2,18, 3,11)( 5,21, 8,16, 6,22, 7,15)( 9,26,20,28,10,25,19,27)(13,31,23,29,14,32,24,30)$
8A-1	$8^{4}$	$4$	$8$	$28$	$( 1,11, 3,18, 2,12, 4,17)( 5,15, 7,22, 6,16, 8,21)( 9,27,19,25,10,28,20,26)(13,30,24,32,14,29,23,31)$
8B	$8^{4}$	$8$	$8$	$28$	$( 1,17, 4,12, 2,18, 3,11)( 5,22, 8,15, 6,21, 7,16)( 9,26,20,28,10,25,19,27)(13,32,23,30,14,31,24,29)$
8C1	$8^{2},4^{4}$	$16$	$8$	$26$	$( 1,25, 4,27, 2,26, 3,28)( 5,23, 6,24)( 7,14, 8,13)( 9,18,20,11,10,17,19,12)(15,30,16,29)(21,32,22,31)$
8C3	$8^{2},4^{4}$	$16$	$8$	$26$	$( 1,10, 2, 9)( 3,19, 4,20)( 5,31, 8,29, 6,32, 7,30)(11,26,12,25)(13,21,23,16,14,22,24,15)(17,27,18,28)$
16A1	$16^{2}$	$16$	$16$	$30$	$( 1,31,17,23, 4,29,12,14, 2,32,18,24, 3,30,11,13)( 5, 9,21,26, 8,20,16,28, 6,10,22,25, 7,19,15,27)$
16A-1	$16^{2}$	$16$	$16$	$30$	$( 1,32,11,23, 3,30,18,14, 2,31,12,24, 4,29,17,13)( 5,10,22,28, 8,20,15,26, 6, 9,21,27, 7,19,16,25)$
16A3	$16^{2}$	$16$	$16$	$30$	$( 1,24,12,31, 3,14,17,30, 2,23,11,32, 4,13,18,29)( 5,25,16, 9, 7,28,21,19, 6,26,15,10, 8,27,22,20)$
16A-3	$16^{2}$	$16$	$16$	$30$	$( 1,29,18,13, 4,32,11,23, 2,30,17,14, 3,31,12,24)( 5,20,22,27, 8,10,15,26, 6,19,21,28, 7, 9,16,25)$

Malle's constant $a(G)$: $1/8$

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	2D	4A	4B	4C	4D1	4D-1	4E	4F1	4F-1	8A1	8A-1	8B	8C1	8C3	16A1	16A-1	16A3	16A-3
	Size	1	1	2	16	16	4	4	4	8	8	16	32	32	4	4	8	16	16	16	16	16	16
	2 P	1A	1A	1A	1A	1A	2B	2A	2B	2B	2B	2A	2D	2D	4B	4B	4B	4C	4C	8A1	8A-1	8A1	8A-1
	Type
256.507.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
256.507.1b	R	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$
256.507.1c	R	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
256.507.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
256.507.1e1	C	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- i$	$i$	$- 1$	$- 1$	$- 1$	$1$	$1$	$i$	$- i$	$- i$	$i$
256.507.1e2	C	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$i$	$- i$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- i$	$i$	$i$	$- i$
256.507.1f1	C	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- i$	$i$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- i$	$i$	$i$	$- i$
256.507.1f2	C	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$i$	$- i$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$i$	$- i$	$- i$	$i$
256.507.2a	R	$2$	$2$	$2$	$0$	$- 2$	$2$	$2$	$2$	$0$	$0$	$- 2$	$0$	$0$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
256.507.2b	R	$2$	$2$	$2$	$0$	$2$	$2$	$2$	$2$	$0$	$0$	$- 2$	$0$	$0$	$- 2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$
256.507.4a	R	$4$	$4$	$4$	$0$	$0$	$- 4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.507.4b	R	$4$	$4$	$4$	$- 2$	$0$	$0$	$- 4$	$0$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.507.4c	R	$4$	$4$	$4$	$2$	$0$	$0$	$- 4$	$0$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.507.4d1	C	$4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$2$	$0$	$0$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$
256.507.4d2	C	$4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$2$	$0$	$0$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$
256.507.4e1	R	$4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$2$	$2$	$- 2$	$0$	$0$	$- ζ_{8}^{- 1} - ζ_{8}$	$- ζ_{8}^{- 1} - ζ_{8}$	$ζ_{8}^{- 1} + ζ_{8}$	$ζ_{8}^{- 1} + ζ_{8}$
256.507.4e2	R	$4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$2$	$2$	$- 2$	$0$	$0$	$ζ_{8}^{- 1} + ζ_{8}$	$ζ_{8}^{- 1} + ζ_{8}$	$- ζ_{8}^{- 1} - ζ_{8}$	$- ζ_{8}^{- 1} - ζ_{8}$
256.507.4f1	C	$4$	$- 4$	$0$	$0$	$0$	$2$	$0$	$- 2$	$- 2 ζ_{8}^{2}$	$2 ζ_{8}^{2}$	$0$	$0$	$0$	$- 2 ζ_{8} - 2 ζ_{8}^{3}$	$2 ζ_{8} + 2 ζ_{8}^{3}$	$0$	$- ζ_{8}^{- 1} - ζ_{8}$	$ζ_{8}^{- 1} + ζ_{8}$	$0$	$0$	$0$	$0$
256.507.4f2	C	$4$	$- 4$	$0$	$0$	$0$	$2$	$0$	$- 2$	$2 ζ_{8}^{2}$	$- 2 ζ_{8}^{2}$	$0$	$0$	$0$	$2 ζ_{8} + 2 ζ_{8}^{3}$	$- 2 ζ_{8} - 2 ζ_{8}^{3}$	$0$	$- ζ_{8}^{- 1} - ζ_{8}$	$ζ_{8}^{- 1} + ζ_{8}$	$0$	$0$	$0$	$0$
256.507.4f3	C	$4$	$- 4$	$0$	$0$	$0$	$2$	$0$	$- 2$	$- 2 ζ_{8}^{2}$	$2 ζ_{8}^{2}$	$0$	$0$	$0$	$2 ζ_{8} + 2 ζ_{8}^{3}$	$- 2 ζ_{8} - 2 ζ_{8}^{3}$	$0$	$ζ_{8}^{- 1} + ζ_{8}$	$- ζ_{8}^{- 1} - ζ_{8}$	$0$	$0$	$0$	$0$
256.507.4f4	C	$4$	$- 4$	$0$	$0$	$0$	$2$	$0$	$- 2$	$2 ζ_{8}^{2}$	$- 2 ζ_{8}^{2}$	$0$	$0$	$0$	$- 2 ζ_{8} - 2 ζ_{8}^{3}$	$2 ζ_{8} + 2 ζ_{8}^{3}$	$0$	$ζ_{8}^{- 1} + ζ_{8}$	$- ζ_{8}^{- 1} - ζ_{8}$	$0$	$0$	$0$	$0$
256.507.8a	R	$8$	$- 8$	$0$	$0$	$0$	$- 4$	$0$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

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	32T7297

Degree	$32$
Order	$256$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$(C_4\times C_8).D_4$