C₂₅×#405 #13098554

order	1600
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₂₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₂₅×#405

Subgroups

	subgroup	G/H
center	C₂×C₈×C₂₅	C₂²
commutator	C₂	C₂×C₄²×C₂₅
Frattini, cosocle	C₂×C₄×C₅	C₂³×C₅
socle	C₂²×C₅	C₂²×C₄×C₅
prime radical	C₂³×C₅	C₂×C₄×C₅
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₅ ◁ G

195 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 195 conjugacy classes of subgroups — too many to draw. Only its 171 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)(30,31,33,36,34,37,40,43)(32,35,39,42,38,41,44,45);(26,27,28,29)(30,32)(31,35)(33,39)(34,38)(36,42)(37,41)(40,44)(43,45);(32,38)(35,41)(39,44)(42,45)

Conjugacy classes & character table

1000 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 906);