#72277300

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
prime radical	C₅⋊#6732	C₂
LCS	G ▷ C₅×SL(2,𝔽₅)
UCS	C₁ ◁ C₂

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

normal subgroups only

This group has 133 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:

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

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