C₅×#7229 #11929949

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

Aliases

direct product

C₅×#7229

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

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

normal subgroups only

This group has 244 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)(6,7,10,15,12,17,24,40)(8,18,20,42,22,44,45,61)(9,23,25,14,28,37,27,39)(11,30,29,41,31,59,53,16)(13,32,33,19,36,46,35,48)(21,50,49,65,51,69,66,43)(26,55,54,52,57,38,56,64)(34,47,62,67,63,68,58,60);(6,8,11,21,24,45,53,66,12,22,31,51,10,20,29,49)(7,13,16,34,15,35,30,58,17,36,41,63,40,33,59,62)(9,19,26,47,25,46,54,67,28,48,57,68,27,32,56,60)(14,18,38,43,23,42,52,50,39,44,55,65,37,61,64,69)

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, 217);