LMFDB - Galois group: 32T506

Show commands: Magma

magma:G := TransitiveGroup(32, 506);

Group invariants

Abstract group:	$C_2^5:C_4$	magma:IdentifyGroup(G);
Order:	$128=2^{7}$	magma:Order(G);
Cyclic:	no	magma:IsCyclic(G);
Abelian:	no	magma:IsAbelian(G);
Solvable:	yes	magma:IsSolvable(G);
Nilpotency class:	$4$	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$32$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$506$	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:	$(1,21,4,23)(2,22,3,24)(5,18,8,20)(6,17,7,19)(9,31,25,16)(10,32,26,15)(11,30,28,13)(12,29,27,14)$, $(1,6,4,7)(2,5,3,8)(9,31,25,16)(10,32,26,15)(11,30,28,13)(12,29,27,14)(17,23,19,21)(18,24,20,22)$, $(1,25)(2,26)(3,27)(4,28)(5,6)(7,8)(9,17)(10,18)(11,19)(12,20)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 7
$4$: $C_4$ x 4, $C_2^2$ x 7
$8$: $D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$16$: $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
$32$: $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$
$64$: $((C_8 : C_2):C_2):C_2$ x 2, 16T76

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_4$ x 4, $C_2^2$ x 7
$8$:	$D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$16$:	$D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
$32$:	$C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$
$64$:	$((C_8 : C_2):C_2):C_2$ x 2, 16T76

Subfields

Degree 2: $C_2$ x 3
Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 4
Degree 8: $C_4\times C_2$, $D_4$ x 2, $C_2^2:C_4$ x 2, $(((C_4 \times C_2): C_2):C_2):C_2$ x 4
Degree 16: $C_2^2 : C_4$, 16T171 x 2, 16T227 x 4

Low degree siblings

16T227 x 4, 16T259 x 8, 16T261 x 4, 16T273 x 4, 16T283 x 4, 32T507 x 2, 32T508 x 4, 32T595 x 4, 32T596 x 8, 32T597 x 2, 32T598 x 4, 32T599 x 4, 32T601, 32T602 x 2, 32T633, 32T657, 32T1130 x 2, 32T1796
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	$2^{16}$	$1$	$2$	$16$	$( 1,17)( 2,18)( 3,20)( 4,19)( 5,24)( 6,23)( 7,21)( 8,22)( 9,25)(10,26)(11,28)(12,27)(13,30)(14,29)(15,32)(16,31)$
2B	$2^{16}$	$1$	$2$	$16$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,32)(30,31)$
2C	$2^{16}$	$1$	$2$	$16$	$( 1,19)( 2,20)( 3,18)( 4,17)( 5,22)( 6,21)( 7,23)( 8,24)( 9,28)(10,27)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)$
2D	$2^{16}$	$2$	$2$	$16$	$( 1,17)( 2,18)( 3,20)( 4,19)( 5, 8)( 6, 7)( 9,25)(10,26)(11,28)(12,27)(13,16)(14,15)(21,23)(22,24)(29,32)(30,31)$
2E	$2^{8},1^{16}$	$2$	$2$	$8$	$( 5,22)( 6,21)( 7,23)( 8,24)(13,31)(14,32)(15,29)(16,30)$
2F	$2^{16}$	$4$	$2$	$16$	$( 1,18)( 2,17)( 3,19)( 4,20)( 5,14)( 6,13)( 7,16)( 8,15)( 9,26)(10,25)(11,27)(12,28)(21,31)(22,32)(23,30)(24,29)$
2G	$2^{16}$	$4$	$2$	$16$	$( 1, 2)( 3, 4)( 5,29)( 6,30)( 7,31)( 8,32)( 9,10)(11,12)(13,23)(14,24)(15,22)(16,21)(17,18)(19,20)(25,26)(27,28)$
2H	$2^{16}$	$4$	$2$	$16$	$( 1,12)( 2,11)( 3, 9)( 4,10)( 5,30)( 6,29)( 7,32)( 8,31)(13,24)(14,23)(15,21)(16,22)(17,27)(18,28)(19,26)(20,25)$
2I	$2^{16}$	$4$	$2$	$16$	$( 1,27)( 2,28)( 3,25)( 4,26)( 5,13)( 6,14)( 7,15)( 8,16)( 9,20)(10,19)(11,18)(12,17)(21,32)(22,31)(23,29)(24,30)$
2J	$2^{16}$	$4$	$2$	$16$	$( 1,28)( 2,27)( 3,26)( 4,25)( 5, 7)( 6, 8)( 9,19)(10,20)(11,17)(12,18)(13,15)(14,16)(21,24)(22,23)(29,31)(30,32)$
2K	$2^{16}$	$4$	$2$	$16$	$( 1,11)( 2,12)( 3,10)( 4, 9)( 5,21)( 6,22)( 7,24)( 8,23)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,25)(20,26)$
2L	$2^{8},1^{16}$	$4$	$2$	$8$	$( 5,22)( 6,21)( 7,23)( 8,24)( 9,28)(10,27)(11,25)(12,26)$
2M	$2^{16}$	$4$	$2$	$16$	$( 1,17)( 2,18)( 3,20)( 4,19)( 5, 8)( 6, 7)( 9,11)(10,12)(13,30)(14,29)(15,32)(16,31)(21,23)(22,24)(25,28)(26,27)$
4A	$4^{8}$	$4$	$4$	$24$	$( 1,26,19,12)( 2,25,20,11)( 3,28,18, 9)( 4,27,17,10)( 5,30,22,16)( 6,29,21,15)( 7,32,23,14)( 8,31,24,13)$
4B	$4^{8}$	$4$	$4$	$24$	$( 1,10,19,27)( 2, 9,20,28)( 3,11,18,25)( 4,12,17,26)( 5,13,22,31)( 6,14,21,32)( 7,15,23,29)( 8,16,24,30)$
4C	$4^{4},2^{8}$	$8$	$4$	$20$	$( 1, 2)( 3, 4)( 5,29,22,15)( 6,30,21,16)( 7,31,23,13)( 8,32,24,14)( 9,27)(10,28)(11,26)(12,25)(17,18)(19,20)$
4D	$4^{4},2^{8}$	$8$	$4$	$20$	$( 1,18)( 2,17)( 3,19)( 4,20)( 5,14,22,32)( 6,13,21,31)( 7,16,23,30)( 8,15,24,29)( 9,12)(10,11)(25,27)(26,28)$
4E1	$4^{8}$	$8$	$4$	$24$	$( 1,14,17,29)( 2,13,18,30)( 3,16,20,31)( 4,15,19,32)( 5,25, 8,28)( 6,26, 7,27)( 9,22,11,24)(10,21,12,23)$
4E-1	$4^{8}$	$8$	$4$	$24$	$( 1,29,17,14)( 2,30,18,13)( 3,31,20,16)( 4,32,19,15)( 5,28, 8,25)( 6,27, 7,26)( 9,24,11,22)(10,23,12,21)$
4F1	$4^{8}$	$8$	$4$	$24$	$( 1,29,17,14)( 2,30,18,13)( 3,31,20,16)( 4,32,19,15)( 5, 9, 8,11)( 6,10, 7,12)(21,27,23,26)(22,28,24,25)$
4F-1	$4^{8}$	$8$	$4$	$24$	$( 1,14,17,29)( 2,13,18,30)( 3,16,20,31)( 4,15,19,32)( 5,11, 8, 9)( 6,12, 7,10)(21,26,23,27)(22,25,24,28)$
8A1	$8^{4}$	$8$	$8$	$28$	$( 1,30,10, 8,19,16,27,24)( 2,29, 9, 7,20,15,28,23)( 3,32,11, 6,18,14,25,21)( 4,31,12, 5,17,13,26,22)$
8A-1	$8^{4}$	$8$	$8$	$28$	$( 1,13,27,22,19,31,10, 5)( 2,14,28,21,20,32, 9, 6)( 3,15,25,23,18,29,11, 7)( 4,16,26,24,17,30,12, 8)$
8B1	$8^{4}$	$8$	$8$	$28$	$( 1,13,10,22,19,31,27, 5)( 2,14, 9,21,20,32,28, 6)( 3,15,11,23,18,29,25, 7)( 4,16,12,24,17,30,26, 8)$
8B-1	$8^{4}$	$8$	$8$	$28$	$( 1,30,27, 8,19,16,10,24)( 2,29,28, 7,20,15, 9,23)( 3,32,25, 6,18,14,11,21)( 4,31,26, 5,17,13,12,22)$

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

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	2L	2M	4A	4B	4C	4D	4E1	4E-1	4F1	4F-1	8A1	8A-1	8B1	8B-1
	Size	1	1	1	1	2	2	4	4	4	4	4	4	4	4	4	4	8	8	8	8	8	8	8	8	8	8
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	2C	2C	2E	2E	2M	2M	2M	2M	4B	4B	4B	4B
	Type
128.850.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
128.850.1b	R	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$
128.850.1c	R	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$
128.850.1d	R	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$
128.850.1e	R	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$
128.850.1f	R	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$
128.850.1g	R	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
128.850.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
128.850.1i1	C	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$- i$	$i$	$- 1$	$1$	$i$	$- i$	$i$	$- i$	$- i$	$i$
128.850.1i2	C	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$i$	$- i$	$- 1$	$1$	$- i$	$i$	$- i$	$i$	$i$	$- i$
128.850.1j1	C	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- i$	$i$	$1$	$- 1$	$i$	$- i$	$- i$	$i$	$i$	$- i$
128.850.1j2	C	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$i$	$- i$	$1$	$- 1$	$- i$	$i$	$i$	$- i$	$- i$	$i$
128.850.1k1	C	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- i$	$i$	$1$	$1$	$- i$	$i$	$i$	$- i$	$i$	$- i$
128.850.1k2	C	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$i$	$- i$	$1$	$1$	$i$	$- i$	$- i$	$i$	$- i$	$i$
128.850.1l1	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- i$	$i$	$- 1$	$- 1$	$- i$	$i$	$- i$	$i$	$- i$	$i$
128.850.1l2	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$i$	$- i$	$- 1$	$- 1$	$i$	$- i$	$i$	$- i$	$i$	$- i$
128.850.2a	R	$2$	$- 2$	$- 2$	$2$	$- 2$	$2$	$0$	$0$	$- 2$	$2$	$- 2$	$2$	$0$	$0$	$- 2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.850.2b	R	$2$	$- 2$	$- 2$	$2$	$- 2$	$2$	$0$	$0$	$- 2$	$2$	$2$	$- 2$	$0$	$0$	$2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.850.2c	R	$2$	$2$	$2$	$2$	$2$	$2$	$0$	$0$	$- 2$	$- 2$	$- 2$	$- 2$	$0$	$0$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.850.2d	R	$2$	$2$	$2$	$2$	$2$	$2$	$0$	$0$	$- 2$	$- 2$	$2$	$2$	$0$	$0$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.850.4a	R	$4$	$- 4$	$- 4$	$4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.850.4b	R	$4$	$4$	$4$	$4$	$- 4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.850.4c	R	$4$	$- 4$	$4$	$- 4$	$0$	$0$	$- 2$	$2$	$0$	$0$	$0$	$0$	$- 2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.850.4d	R	$4$	$- 4$	$4$	$- 4$	$0$	$0$	$2$	$- 2$	$0$	$0$	$0$	$0$	$2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.850.4e	R	$4$	$4$	$- 4$	$- 4$	$0$	$0$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.850.4f	R	$4$	$4$	$- 4$	$- 4$	$0$	$0$	$2$	$2$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

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	32T506

Degree	$32$
Order	$128$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$C_2^5:C_4$