C₅×#13495 #11936215

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

Aliases

direct product

C₅×#13495

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)(14,15,17,22,30,40,44,39,19,24,31,38,18,23,29,21)(16,20,26,33,27,34,43,45,28,36,42,32,25,35,41,37);(7,10)(11,13)(14,16)(15,20)(17,26)(18,25)(19,28)(21,37)(22,33)(23,35)(24,36)(27,30)(29,41)(31,42)(32,38)(34,40)(39,45)(43,44);(6,8,9,12)(7,11,10,13)(15,21)(16,25)(17,29)(18,30)(20,32)(22,23)(24,39)(26,42)(27,28)(31,44)(33,36)(34,45)(35,37)(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, 6483);