C₂₅×#419 #13098568

order	1600
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₅×#419

Subgroups

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

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

normal subgroups only

This group has 225 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)(34,43)(35,38)(36,41)(40,44)(42,45);(30,32,35,42)(31,36,38,44)(33,41,34,40)(37,45,43,39);(30,33,35,34)(31,37,38,43)(32,40,42,41)(36,39,44,45)

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, 920);