#72268759

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#72209728
commutator, hypocenter	G	C₁
Frattini, socle	C₂⁶	#11757163
Fitting, cosocle	C₂×#153709	A₅
LCS	G
UCS	C₁ ◁ C₂²

3030 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has 3030 conjugacy classes of subgroups — too many to draw. Only its 13 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) = 72
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 72 points:

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

Conjugacy classes & character table

102 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((1,19,26,30,15,3,29,8)(2,14,12,18,23,24,11,27)(4,21,9,10,22,5,25,6)(7,13,28,17,32,31,16,20)(33,58,54,63,37,49,41,60)(34,67)(35,61,53,57,43,68,65,42)(36,55)(38,70,48,66,44,71,47,72)(39,45)(40,69)(46,59)(50,62,56,64)(51,52), (1,2,5,3,9,24)(4,12)(6,17,29,18,7,8)(10,20,26,27,32,30)(11,22)(13,28)(14,15,23,21,19,25)(16,31)(33,34,38,37,36,44)(35,39,49,53,51,63,43,45,58,65,52,60)(40,47,41,46,48,54)(42,55,62,57,67,64)(50,61,69,56,68,59)(66,70,72,71));