#72240076

order	61440
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₂	#72210011
commutator, hypocenter	G	C₁
Frattini	C₂⁴×C₄	#11757162
Fitting, cosocle	C₄⁵	A₅
LCS	G
UCS	C₁ ◁ C₂

2600 conjugacy classes of subgroups; only the 6 normal subgroups are shown.

normal subgroups only

This group has 2600 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) = 48
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 48 points:

(1,23,25,34,6,14,28,18,27,10,39,4)(2,42,45,8,29,40,24,36,48,20,15,31)(3,19,22,16,26,5,47,46,30,12,41,11)(7,43,35,21,33,38,37,32,17,9,13,44);(1,25,10,4,28,27,34,14)(2,7,20,9,24,37,8,21)(3,22,12,11,47,30,16,5)(6,35)(13,36,32,29,33,42,43,15)(17,39)(18,44)(19,31)(23,38)(26,48)(40,46)(41,45)

Conjugacy classes & character table

88 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((1,23,25,34,6,14,28,18,27,10,39,4)(2,42,45,8,29,40,24,36,48,20,15,31)(3,19,22,16,26,5,47,46,30,12,41,11)(7,43,35,21,33,38,37,32,17,9,13,44), (1,25,10,4,28,27,34,14)(2,7,20,9,24,37,8,21)(3,22,12,11,47,30,16,5)(6,35)(13,36,32,29,33,42,43,15)(17,39)(18,44)(19,31)(23,38)(26,48)(40,46)(41,45));