C₅×#13450 #11936170

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13450

Subgroups

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

410 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 1280

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

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

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6438);