C₅×#7227 #11929947

order	1280
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7227

Subgroups

	subgroup	G/H
center	C₄×C₅	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₅
Frattini, cosocle	C₂×#131	C₂²×C₅
socle	C₂×C₅	#1289
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈ ◁ G

248 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 69 points:

(1,2,3,4,5)(6,7)(8,18)(9,23,25,49)(10,28)(11,16)(12,17)(13,19,31,43)(14,27,34,51)(15,36)(20,46)(21,40)(22,41)(24,37)(26,53,52,35)(29,38)(30,56)(32,59,57,67)(33,45,58,64)(39,60)(42,62)(44,66,65,55)(47,63)(48,68)(50,54,69,61);(6,8,10,20,11,21,29,47,12,22,30,48,24,42,36,60)(7,13,15,32,37,58,56,66,17,31,38,57,16,33,28,55)(9,19,26,44,27,45,54,67,25,43,52,65,51,64,61,59)(14,18,35,39,49,62,69,68,34,41,53,63,23,40,50,46)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 215);