C₅×#10666 #11933386

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10666

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#399
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

354 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 320

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

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

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, 3654);