C₅×#13496 #11936216

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13496

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

284 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 45
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,20,32,31,42,30,22,19,25,17,23,18,24,34,41)(16,21,33,35,28,37,43,40,29,39,27,36,26,38,44,45);(7,10)(11,13)(14,16)(15,21)(17,27)(18,26)(19,29)(20,33)(22,40)(23,36)(24,38)(25,39)(28,31)(30,43)(32,35)(34,44)(37,42)(41,45);(6,8,9,12)(7,11,10,13)(15,22)(16,26)(17,30)(18,31)(20,34)(21,35)(23,24)(25,41)(27,33)(28,29)(32,42)(36,39)(37,45)(38,40)

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, 6484);