C₅×#12434 #11935154

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12433
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12434

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂×C₅×D₄	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈×D₄ ◁ G

230 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 82 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)(8,10)(9,12)(11,13)(14,15,19,23)(16,24,26,38)(17,29,31,44)(18,32,30,22)(20,33,34,27)(21,36,37,25)(28,43,42,40)(35,41,39,45);(14,16,25,41,18,28,31,33,19,26,36,45,30,42,17,27)(15,20,29,40,22,35,37,38,23,34,44,43,32,39,21,24);(7,10)(8,13)(11,12)(16,26)(20,34)(24,38)(27,33)(28,42)(35,39)(40,43)(41,45)

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5422);