D₅×#226⋊A₄ #72277293

order	5760
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

D₅×#226⋊A₄

Splittings

Sylow

C₅ ⋊ C₂×#226⋊A₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅×A₄×S₄
commutator, cosocle, hypocenter	C₅×#994	C₂²×C₃
Fitting	C₅×H(5,𝔽₂)	C₂×C₃×D₃
socle	C₂×C₅	C₂×A₄×S₄
LCS	G ▷ C₅×#994
UCS	C₁ ◁ C₂

790 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 43 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) = 53
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 2 generators on 53 points:

(2,3)(4,5)(6,9,12,13,25,10)(7,49,36,18,45,34)(8,21,19,22,43,26)(11,31,30,32,28,46)(14,17,48,37,39,51)(15,47,38,40,52,41)(16,50,53,33,29,35)(20,44)(23,27)(24,42);(1,2,4,5,3)(6,39)(7,27)(8,38)(9,33)(10,15)(11,51)(12,40)(13,17)(14,43)(16,25)(18,23)(19,53)(20,34)(21,37)(22,41)(24,52)(26,35)(28,45)(29,46)(30,50)(31,49)(32,48)(36,44)(42,47)

Conjugacy classes & character table

116 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((2,3)(4,5)(6,9,12,13,25,10)(7,49,36,18,45,34)(8,21,19,22,43,26)(11,31,30,32,28,46)(14,17,48,37,39,51)(15,47,38,40,52,41)(16,50,53,33,29,35)(20,44)(23,27)(24,42), (1,2,4,5,3)(6,39)(7,27)(8,38)(9,33)(10,15)(11,51)(12,40)(13,17)(14,43)(16,25)(18,23)(19,53)(20,34)(21,37)(22,41)(24,52)(26,35)(28,45)(29,46)(30,50)(31,49)(32,48)(36,44)(42,47));