#72277246

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

Aliases

No aliases for this group.

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

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

normal subgroups only

This group has 139 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) = 53
transitive degree	μ_t(G) = 240

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

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277251, #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)(6,27,8,34)(7,40,14,45)(9,39,18,16)(10,36,12,46)(11,42,19,51)(13,22,35,17)(15,26,38,33)(20,23,47,30)(21,25,29,32)(24,48,31,28)(37,43,52,49)(41,50,44,53), (1,2)(4,5)(6,16,43,30,7,29,51,10,31,53,8,39,49,23,14,21,42,12,24,50)(9,32,38,41,20,11,34,22,45,28,18,25,15,44,47,19,27,17,40,48)(13,33,35,26)(36,37,46,52));