C₂₅×#418 #13098567

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

Aliases

direct product

C₂₅×#418

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	C₂²×C₅×D₄	C₂×C₅
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅ ◁ C₂²×C₄×C₂₅ ◁ G

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

normal subgroups only

This group has 285 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) = 37
transitive degree	μ_t(G) = 400

A minimal faithful permutation representation — 3 generators on 37 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)(32,36)(33,35)(34,37);(31,35)(32,33)(36,37);(30,32,34,33)(31,36,37,35)

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, 919);