C₂₅×#443 #13098592

order	1600
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₅×#443

Subgroups

	subgroup	G/H
center	C₈×C₂₅	D₄
commutator	C₄	C₂²×C₄×C₂₅
Frattini, cosocle	C₂×C₄×C₅	C₂³×C₅
socle	C₂×C₅	C₄×C₅×D₄
prime radical	C₅×#135	C₂×C₅
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₅ ◁ C₂×C₈×C₂₅ ◁ G

159 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 159 conjugacy classes of subgroups — too many to draw. Only its 99 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)(26,27,30,34,31,28,35,33)(29,32,37,40,38,36,41,39);(26,28)(27,31)(29,36)(30,33)(32,38)(34,35)(37,39)(40,41);(26,29,30,37,31,38,35,41)(27,32,34,40,28,36,33,39)

Conjugacy classes & character table

700 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 944);