C₂₅×#410 #13098559

order	1600
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂₅×#410

Subgroups

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

237 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 117 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) = 41
transitive degree	μ_t(G) = 400

A minimal faithful permutation representation — 3 generators on 41 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)(28,34,36,31)(29,32)(30,33)(35,40,41,38)(37,39);(26,28)(27,31)(29,35)(30,36)(32,38)(33,34)(37,41)(39,40);(26,29,30,37)(27,32,33,39)(28,35,36,41)(31,38,34,40)

Conjugacy classes & character table

550 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 911);