C₅×#13306 #11936026

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13306

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂⁴×C₅	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

566 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 138 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)(8,9)(14,15,19,23)(16,24,27,43)(17,28,32,42)(18,33,30,22)(20,35,37,25)(21,38,40,31)(26,44,34,39)(29,41,45,36);(10,11)(12,13)(14,16,19,27)(15,20,23,37)(17,29,32,45)(18,34,30,26)(21,39,40,44)(22,41,33,36)(24,42,43,28)(25,38,35,31);(7,9)(11,13)

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, 6294);