C₂₅×#416 #13098565

order	1600
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₅×#416

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₅	D₄
commutator	C₄	C₂²×C₄×C₂₅
Frattini, cosocle	C₂×C₄×C₅	C₂³×C₅
socle	C₂²×C₅	C₂×C₅×D₄
prime radical	C₂×C₅×#41	C₂×C₅
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₅ ◁ C₂²×C₄×C₂₅ ◁ G

237 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 45
transitive degree	μ_t(G) = 800

A minimal faithful permutation representation — 3 generators on 45 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)(26,27,28,29)(30,31)(32,39)(33,37)(34,42)(35,38)(36,40)(41,44)(43,45);(30,32)(31,36)(33,40)(34,43)(35,41)(37,44)(38,45)(39,42);(30,33,35,34)(31,37,38,42)(32,40,41,43)(36,44,45,39)

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, 917);