C₅×#10676 #11933396

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#10676

Subgroups

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

410 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 37
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 37 points:

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

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, 3664);