C₅×#13512 #11936232

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13512

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2085
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

232 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 102 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)(6,7,12,16)(8,17,14,10)(9,18)(11,26,28,15)(13,23)(19,25,33,34)(20,29)(21,32)(22,35)(24,30)(27,37)(31,36);(6,8,10,7,12,14,17,16)(9,20,21,31,24,35,23,37)(11,26,25,19,28,15,34,33)(13,29,30,27,32,22,18,36);(6,9,10,21,12,24,17,23)(7,13,14,30,16,32,8,18)(11,27,25,22,28,36,34,29)(15,35,33,37,26,20,19,31)

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, 6500);