C₅×#13504 #11936224

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13504

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2077
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

278 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 98 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)(6,7,14,30,12,17,11,18,13,20,28,36,10,19,29,31)(8,21,26,33,22,15,23,16,24,35,37,32,9,25,27,34);(6,8)(7,15)(9,10)(11,23)(12,22)(13,24)(14,26)(16,30)(17,35)(18,32)(19,21)(20,25)(27,29)(28,37)(31,33)(34,36);(6,9,10,8,13,22,12,24)(7,16,17,33,20,34,19,32)(11,27,14,26,29,23,28,37)(15,31,21,30,25,18,35,36)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6492);