C₂₅×#473 #13098622

order	1600
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₅×#473

Subgroups

	subgroup	G/H
center	C₂×C₂₅	#121
commutator	C₂×C₄	C₂³×C₂₅
Frattini, cosocle	C₂×C₄×C₅	C₂³×C₅
socle	C₂×C₅	C₅×#121
prime radical	C₅×#41	C₂²×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₅ ◁ C₂×C₄×C₂₅ ◁ G

150 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 57
transitive degree	μ_t(G) = 800

A minimal faithful permutation representation — 3 generators on 57 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(26,27,32,37)(28,38,42,48)(29,43,45,51)(30,36,31,35)(33,41,49,40)(34,46,50,44)(39,53,55,47)(52,56,57,54);(26,28,31,41,32,42,30,40)(27,33,36,48,37,49,35,38)(29,39,46,56,45,55,44,54)(34,47,51,57,50,53,43,52);(26,29)(27,34)(28,39)(30,44)(31,46)(32,45)(33,47)(35,43)(36,51)(37,50)(38,52)(40,54)(41,56)(42,55)(48,57)(49,53)

Conjugacy classes & character table

400 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 974);