Finite Group Databasebeta

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#72209943

order	29120
isoclinism	isoclinic class of #72209943
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	G	C₁
LCS	G
UCS	C₁

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

(1,2)(3,7)(4,9)(5,10)(6,12)(8,15)(11,17)(13,20)(14,22)(16,24)(18,27)(19,26)(21,29)(23,32)(25,35)(28,36)(30,40)(31,42)(33,44)(34,46)(37,41)(38,49)(39,50)(43,54)(45,55)(47,57)(48,52)(51,59)(53,61)(56,62)(60,64)(63,65);(1,3,8,4)(2,5,11,6)(7,13,21,14)(10,15,23,16)(12,18,28,19)(17,25,20,26)(22,30,41,31)(24,33,45,34)(27,36,48,37)(29,38,44,39)(32,42,53,43)(35,46,56,47)(40,51,60,52)(49,57,65,58)(50,54,62,59)(55,63,61,64)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁
size	1	455	1820	1820	2240	2240	2240	4160	4160	4160	5824
order	1	2	4	4	13	13	13	7	7	7	5
χ₁	1	1	1	1	1	1	1	1	1	1	1
χ₂	14	-2	$2 i$	$- 2 i$	1	1	1	0	0	0	-1
χ₃	14	-2	$- 2 i$	$2 i$	1	1	1	0	0	0	-1
χ₄	35	3	-1	-1	−ζ⁶−ζ²²−ζ³⁰−ζ⁴⁶−ζ⁵⁸−ζ⁷⁴−ζ⁸²−ζ¹¹⁰−ζ¹³⁴−ζ¹⁵⁰−ζ¹⁶²−ζ¹⁷⁸−ζ¹⁸⁶−ζ²⁰²−ζ²¹⁴−ζ²³⁰−ζ²⁵⁴−ζ²⁸²−ζ²⁹⁰−ζ³⁰⁶−ζ³¹⁸−ζ³³⁴−ζ³⁴²−ζ³⁵⁸	−ζ¹⁸−ζ³⁴−ζ³⁸−ζ⁶⁶−ζ⁸⁶−ζ⁹⁰−ζ¹¹⁸−ζ¹²²−ζ¹³⁸−ζ¹⁴²−ζ¹⁷⁰−ζ¹⁷⁴−ζ¹⁹⁰−ζ¹⁹⁴−ζ²²²−ζ²²⁶−ζ²⁴²−ζ²⁴⁶−ζ²⁷⁴−ζ²⁷⁸−ζ²⁹⁸−ζ³²⁶−ζ³³⁰−ζ³⁴⁶	−ζ²−ζ¹⁰−ζ⁵⁰−ζ⁵⁴−ζ⁶²−ζ⁹⁴−ζ¹⁰²−ζ¹⁰⁶−ζ¹¹⁴−ζ¹⁴⁶−ζ¹⁵⁸−ζ¹⁶⁶−ζ¹⁹⁸−ζ²⁰⁶−ζ²¹⁸−ζ²⁵⁰−ζ²⁵⁸−ζ²⁶²−ζ²⁷⁰−ζ³⁰²−ζ³¹⁰−ζ³¹⁴−ζ³⁵⁴−ζ³⁶²	0	0	0	0
χ₅	35	3	-1	-1	−ζ¹⁸−ζ³⁴−ζ³⁸−ζ⁶⁶−ζ⁸⁶−ζ⁹⁰−ζ¹¹⁸−ζ¹²²−ζ¹³⁸−ζ¹⁴²−ζ¹⁷⁰−ζ¹⁷⁴−ζ¹⁹⁰−ζ¹⁹⁴−ζ²²²−ζ²²⁶−ζ²⁴²−ζ²⁴⁶−ζ²⁷⁴−ζ²⁷⁸−ζ²⁹⁸−ζ³²⁶−ζ³³⁰−ζ³⁴⁶	−ζ²−ζ¹⁰−ζ⁵⁰−ζ⁵⁴−ζ⁶²−ζ⁹⁴−ζ¹⁰²−ζ¹⁰⁶−ζ¹¹⁴−ζ¹⁴⁶−ζ¹⁵⁸−ζ¹⁶⁶−ζ¹⁹⁸−ζ²⁰⁶−ζ²¹⁸−ζ²⁵⁰−ζ²⁵⁸−ζ²⁶²−ζ²⁷⁰−ζ³⁰²−ζ³¹⁰−ζ³¹⁴−ζ³⁵⁴−ζ³⁶²	−ζ⁶−ζ²²−ζ³⁰−ζ⁴⁶−ζ⁵⁸−ζ⁷⁴−ζ⁸²−ζ¹¹⁰−ζ¹³⁴−ζ¹⁵⁰−ζ¹⁶²−ζ¹⁷⁸−ζ¹⁸⁶−ζ²⁰²−ζ²¹⁴−ζ²³⁰−ζ²⁵⁴−ζ²⁸²−ζ²⁹⁰−ζ³⁰⁶−ζ³¹⁸−ζ³³⁴−ζ³⁴²−ζ³⁵⁸	0	0	0	0
χ₆	35	3	-1	-1	−ζ²−ζ¹⁰−ζ⁵⁰−ζ⁵⁴−ζ⁶²−ζ⁹⁴−ζ¹⁰²−ζ¹⁰⁶−ζ¹¹⁴−ζ¹⁴⁶−ζ¹⁵⁸−ζ¹⁶⁶−ζ¹⁹⁸−ζ²⁰⁶−ζ²¹⁸−ζ²⁵⁰−ζ²⁵⁸−ζ²⁶²−ζ²⁷⁰−ζ³⁰²−ζ³¹⁰−ζ³¹⁴−ζ³⁵⁴−ζ³⁶²	−ζ⁶−ζ²²−ζ³⁰−ζ⁴⁶−ζ⁵⁸−ζ⁷⁴−ζ⁸²−ζ¹¹⁰−ζ¹³⁴−ζ¹⁵⁰−ζ¹⁶²−ζ¹⁷⁸−ζ¹⁸⁶−ζ²⁰²−ζ²¹⁴−ζ²³⁰−ζ²⁵⁴−ζ²⁸²−ζ²⁹⁰−ζ³⁰⁶−ζ³¹⁸−ζ³³⁴−ζ³⁴²−ζ³⁵⁸	−ζ¹⁸−ζ³⁴−ζ³⁸−ζ⁶⁶−ζ⁸⁶−ζ⁹⁰−ζ¹¹⁸−ζ¹²²−ζ¹³⁸−ζ¹⁴²−ζ¹⁷⁰−ζ¹⁷⁴−ζ¹⁹⁰−ζ¹⁹⁴−ζ²²²−ζ²²⁶−ζ²⁴²−ζ²⁴⁶−ζ²⁷⁴−ζ²⁷⁸−ζ²⁹⁸−ζ³²⁶−ζ³³⁰−ζ³⁴⁶	0	0	0	0
χ₇	64	0	0	0	-1	-1	-1	1	1	1	-1
χ₈	65	1	1	1	0	0	0	ζ¹⁰+ζ¹⁸+ζ³⁸+ζ⁴⁶+ζ⁶⁶+ζ⁷⁴+ζ⁹⁴+ζ¹⁰²+ζ¹²²+ζ¹⁵⁰+ζ¹⁵⁸+ζ¹⁷⁸+ζ¹⁸⁶+ζ²⁰⁶+ζ²¹⁴+ζ²⁴²+ζ²⁶²+ζ²⁷⁰+ζ²⁹⁰+ζ²⁹⁸+ζ³¹⁸+ζ³²⁶+ζ³⁴⁶+ζ³⁵⁴	ζ⁶+ζ²²+ζ³⁴+ζ⁵⁰+ζ⁶²+ζ⁹⁰+ζ¹⁰⁶+ζ¹¹⁸+ζ¹³⁴+ζ¹⁴⁶+ζ¹⁶²+ζ¹⁷⁴+ζ¹⁹⁰+ζ²⁰²+ζ²¹⁸+ζ²³⁰+ζ²⁴⁶+ζ²⁵⁸+ζ²⁷⁴+ζ³⁰²+ζ³¹⁴+ζ³³⁰+ζ³⁴²+ζ³⁵⁸	ζ²+ζ³⁰+ζ⁵⁴+ζ⁵⁸+ζ⁸²+ζ⁸⁶+ζ¹¹⁰+ζ¹¹⁴+ζ¹³⁸+ζ¹⁴²+ζ¹⁶⁶+ζ¹⁷⁰+ζ¹⁹⁴+ζ¹⁹⁸+ζ²²²+ζ²²⁶+ζ²⁵⁰+ζ²⁵⁴+ζ²⁷⁸+ζ²⁸²+ζ³⁰⁶+ζ³¹⁰+ζ³³⁴+ζ³⁶²	0
χ₉	65	1	1	1	0	0	0	ζ⁶+ζ²²+ζ³⁴+ζ⁵⁰+ζ⁶²+ζ⁹⁰+ζ¹⁰⁶+ζ¹¹⁸+ζ¹³⁴+ζ¹⁴⁶+ζ¹⁶²+ζ¹⁷⁴+ζ¹⁹⁰+ζ²⁰²+ζ²¹⁸+ζ²³⁰+ζ²⁴⁶+ζ²⁵⁸+ζ²⁷⁴+ζ³⁰²+ζ³¹⁴+ζ³³⁰+ζ³⁴²+ζ³⁵⁸	ζ²+ζ³⁰+ζ⁵⁴+ζ⁵⁸+ζ⁸²+ζ⁸⁶+ζ¹¹⁰+ζ¹¹⁴+ζ¹³⁸+ζ¹⁴²+ζ¹⁶⁶+ζ¹⁷⁰+ζ¹⁹⁴+ζ¹⁹⁸+ζ²²²+ζ²²⁶+ζ²⁵⁰+ζ²⁵⁴+ζ²⁷⁸+ζ²⁸²+ζ³⁰⁶+ζ³¹⁰+ζ³³⁴+ζ³⁶²	ζ¹⁰+ζ¹⁸+ζ³⁸+ζ⁴⁶+ζ⁶⁶+ζ⁷⁴+ζ⁹⁴+ζ¹⁰²+ζ¹²²+ζ¹⁵⁰+ζ¹⁵⁸+ζ¹⁷⁸+ζ¹⁸⁶+ζ²⁰⁶+ζ²¹⁴+ζ²⁴²+ζ²⁶²+ζ²⁷⁰+ζ²⁹⁰+ζ²⁹⁸+ζ³¹⁸+ζ³²⁶+ζ³⁴⁶+ζ³⁵⁴	0
χ₁₀	65	1	1	1	0	0	0	ζ²+ζ³⁰+ζ⁵⁴+ζ⁵⁸+ζ⁸²+ζ⁸⁶+ζ¹¹⁰+ζ¹¹⁴+ζ¹³⁸+ζ¹⁴²+ζ¹⁶⁶+ζ¹⁷⁰+ζ¹⁹⁴+ζ¹⁹⁸+ζ²²²+ζ²²⁶+ζ²⁵⁰+ζ²⁵⁴+ζ²⁷⁸+ζ²⁸²+ζ³⁰⁶+ζ³¹⁰+ζ³³⁴+ζ³⁶²	ζ¹⁰+ζ¹⁸+ζ³⁸+ζ⁴⁶+ζ⁶⁶+ζ⁷⁴+ζ⁹⁴+ζ¹⁰²+ζ¹²²+ζ¹⁵⁰+ζ¹⁵⁸+ζ¹⁷⁸+ζ¹⁸⁶+ζ²⁰⁶+ζ²¹⁴+ζ²⁴²+ζ²⁶²+ζ²⁷⁰+ζ²⁹⁰+ζ²⁹⁸+ζ³¹⁸+ζ³²⁶+ζ³⁴⁶+ζ³⁵⁴	ζ⁶+ζ²²+ζ³⁴+ζ⁵⁰+ζ⁶²+ζ⁹⁰+ζ¹⁰⁶+ζ¹¹⁸+ζ¹³⁴+ζ¹⁴⁶+ζ¹⁶²+ζ¹⁷⁴+ζ¹⁹⁰+ζ²⁰²+ζ²¹⁸+ζ²³⁰+ζ²⁴⁶+ζ²⁵⁸+ζ²⁷⁴+ζ³⁰²+ζ³¹⁴+ζ³³⁰+ζ³⁴²+ζ³⁵⁸	0
χ₁₁	91	-5	-1	-1	0	0	0	0	0	0	1

Here ζ is a primitive root of unity of order 364 (ζ³⁶⁴ = 1); i = ζ⁹¹.

Relationships

maximal quotient of

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

gap> G := Group((1,2)(3,7)(4,9)(5,10)(6,12)(8,15)(11,17)(13,20)(14,22)(16,24)(18,27)(19,26)(21,29)(23,32)(25,35)(28,36)(30,40)(31,42)(33,44)(34,46)(37,41)(38,49)(39,50)(43,54)(45,55)(47,57)(48,52)(51,59)(53,61)(56,62)(60,64)(63,65), (1,3,8,4)(2,5,11,6)(7,13,21,14)(10,15,23,16)(12,18,28,19)(17,25,20,26)(22,30,41,31)(24,33,45,34)(27,36,48,37)(29,38,44,39)(32,42,53,43)(35,46,56,47)(40,51,60,52)(49,57,65,58)(50,54,62,59)(55,63,61,64));