#72238920

order	3840
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

derived

#14515301 ⋊ C₂

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₂	#14515297
commutator, hypocenter	#14515301	C₂
Frattini, cosocle	C₂⁵	C₂×A₅
Fitting	C₂⁶	A₅
LCS	G ▷ #14515301
UCS	C₁ ◁ C₂

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

normal subgroups only

This group has 470 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) = 24
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 24 points:

(1,16,20)(2,3,8,9,12,18)(4,19,11)(5,23,10)(6,13,14,17,7,22)(15,24,21);(1,15,5,4)(2,8,9,18)(3,11)(6,13,17,7)(10,14)(12,21)(16,24,23,19)(20,22)

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72238921, #72267881

GAP

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

gap> G := Group((1,16,20)(2,3,8,9,12,18)(4,19,11)(5,23,10)(6,13,14,17,7,22)(15,24,21), (1,15,5,4)(2,8,9,18)(3,11)(6,13,17,7)(10,14)(12,21)(16,24,23,19)(20,22));