C₅×#13361 #11936081

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13360
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13361

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₅
prime radical	C₅×#3021	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

686 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 160

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6349);