C₅×#10645 #11933365

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#10645

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 258 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) = 37
transitive degree	μ_t(G) = 640

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

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3633);