C₅×#91881 #72277315

order	2400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

non-solvable

Aliases

direct product

C₅×#91881

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₂²×A₅
commutator, hypocenter	SL(2,𝔽₅)	C₂²×C₅
Frattini, cosocle	C₂	C₂²×C₅×A₅
Fitting, hypercenter	C₅×D₄	A₅
LCS	G ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂×C₅ ◁ C₅×D₄

182 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 22 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:

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

Conjugacy classes & character table

120 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((6,52)(7,49)(8,39)(9,22)(10,17)(11,36)(12,51)(13,14)(15,20)(16,44)(18,28)(19,41)(21,33)(23,46)(24,34)(25,50)(26,47)(27,35)(29,53)(30,37)(31,42)(32,48)(38,40)(43,45), (1,2,3,4,5)(6,7,32,10,14)(8,11,23,18,25)(9,30,50,39,31,19,13,53,20,49)(12,27,34,29,15)(17,28,44,51,45,33,24,47,36,48)(21,37,38,42,43)(35,46)(40,52));