#72277297

order	2400
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

derived

C₅×SL(2,𝔽₅) ⋊ C₂²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅×S₅
commutator, cosocle, hypocenter	C₅×SL(2,𝔽₅)	C₂²
Fitting, socle	C₂×C₅	C₂×S₅
LCS	G ▷ C₅×SL(2,𝔽₅)
UCS	C₁ ◁ C₂

157 conjugacy classes of subgroups; only the 12 normal subgroups are shown.

normal subgroups only

This group has 157 conjugacy classes of subgroups — too many to draw. Only its 12 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) = 45
transitive degree	μ_t(G) = 200

A minimal faithful permutation representation — 2 generators on 45 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277310
maximal normal subgroup of	#72277310

GAP

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

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