#72239028

order	61440
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting

C₂×#153709 ⋊ A₅

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₂	#72209997
commutator, hypocenter	G	C₁
Frattini	C₂⁶	#11757162
Fitting, cosocle	C₂×#153709	A₅
LCS	G
UCS	C₁ ◁ C₂

7493 conjugacy classes of subgroups; only the 6 normal subgroups are shown.

normal subgroups only

This group has 7493 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:

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

Conjugacy classes & character table

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