C₂²×#72209725 #72238332

order	10240
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×#72209725

Splittings

Fitting	C₂²×C₄⁴ ⋊ D₅
derived	C₄⁴⋊C₅ ⋊ C₂³

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#72209725
commutator, cosocle, hypocenter	C₄⁴⋊C₅	C₂³
Frattini	C₂⁴	C₂²×(C₂⁴⋊C₅)⋊C₂
Fitting	C₂²×C₄⁴	D₅
socle	C₂⁶	(C₂⁴⋊C₅)⋊C₂
LCS	G ▷ C₄⁴⋊C₅
UCS	C₁ ◁ C₂²

32664 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 32664 conjugacy classes of subgroups — too many to draw. Only its 31 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) = 22
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 3 generators on 22 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72238336, #72258131

GAP

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

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