C₅×#12403 #11935123

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12403

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂×C₅×D₄	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#436 ◁ G

280 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 640

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7,11,15)(8,16,19,20)(9,14,10,13)(12,18,21,17)(23,28)(24,27)(25,32)(29,36)(30,37)(31,34)(33,35);(6,8,11,19)(7,12,15,21)(9,18,10,17)(13,20,14,16)(22,23,26,30)(24,29,27,35)(25,33,34,36)(28,31,37,32);(7,13)(9,10)(12,20)(14,15)(16,21)(17,18)(22,24,26,27)(23,29,30,35)(25,32,34,31)(28,36,37,33)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5391);