#72237403

order	61440
isoclinism	isoclinic class of #72237403
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting

#72234510 ⋊ A₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	G	C₁
Frattini	C₂⁶	#11757162
Fitting, cosocle	#72234510	A₅
socle	C₂⁴	#72209602
LCS	G
UCS	C₁

At least 20782 conjugacy classes of subgroups; only the 8 normal subgroups are shown.

normal subgroups only

This group has at least 20782 conjugacy classes of subgroups — too many to draw. Only its 8 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:

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

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

GAP

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

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