C₂₅×#456 #13098605

order	1600
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₅×#456

Subgroups

	subgroup	G/H
center	C₂×C₂₅	#120
commutator	C₂×C₄	C₂³×C₂₅
Frattini, cosocle	C₂×C₄×C₅	C₂³×C₅
socle	C₂×C₅	C₅×#120
prime radical	C₅×2₋¹⁺⁴	C₂×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₅ ◁ C₂×C₄×C₂₅ ◁ G

213 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 81 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)(26,27,31,29)(28,34,37,41)(30,38,40,33)(32,39,35,36);(26,28)(27,32)(29,35)(30,39)(31,37)(33,41)(34,38)(36,40);(26,29,31,27)(28,35,37,32)(30,33,40,38)(34,39,41,36)

Conjugacy classes & character table

400 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 957);