#72227521

order	30720
isoclinism	isoclinic class of #72227521
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	G	C₁
Frattini	C₂⁵	#11757163
Fitting, cosocle	#6342371	A₅
socle	C₂⁴	#14515304
LCS	G
UCS	C₁

4098 conjugacy classes of subgroups; only the 6 normal subgroups are shown.

normal subgroups only

This group has 4098 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 2 generators on 80 points:

(1,6,9,8,4,7,3,2)(5,10,12,16,14,18,13,17)(11,19)(15,32,25,33,27,38,26,23)(20,31,21,29)(22,30)(24,44)(28,39,40,48)(34,35,45,36)(37,46)(41,50)(42,49)(43,61,54,51,53,65,55,62)(47,63,59,66,58,52,57,64)(56,71)(60,67)(68,73,69,72)(70,78,77,80,76,74,75,79);(1,5,15,19,2)(3,13,26,29,7)(4,14,27,30,8)(6,9,12,25,31)(10,11,28,47,34)(16,21,41,59,44)(17,20,42,57,46)(18,22,40,58,45)(23,24,43,60,39)(32,35,55,73,49)(33,37,53,71,48)(36,54,72,50,38)(51,52,56,70,74)(61,66,69,77,79)(62,63,67,76,78)(64,68,75,80,65)

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72268932

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,6,9,8,4,7,3,2)(5,10,12,16,14,18,13,17)(11,19)(15,32,25,33,27,38,26,23)(20,31,21,29)(22,30)(24,44)(28,39,40,48)(34,35,45,36)(37,46)(41,50)(42,49)(43,61,54,51,53,65,55,62)(47,63,59,66,58,52,57,64)(56,71)(60,67)(68,73,69,72)(70,78,77,80,76,74,75,79), (1,5,15,19,2)(3,13,26,29,7)(4,14,27,30,8)(6,9,12,25,31)(10,11,28,47,34)(16,21,41,59,44)(17,20,42,57,46)(18,22,40,58,45)(23,24,43,60,39)(32,35,55,73,49)(33,37,53,71,48)(36,54,72,50,38)(51,52,56,70,74)(61,66,69,77,79)(62,63,67,76,78)(64,68,75,80,65));