C₅×#13545 #11936265

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13545
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13545

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2101
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

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

normal subgroups only

This group has 216 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)(6,7,10,15,11,13,23,30)(8,16,26,29,18,12,20,33)(9,21,24,37,25,14,22,32)(17,31,19,28,35,27,36,34);(6,8)(7,12,13,16)(9,17)(10,26)(11,18)(14,31,21,27)(15,33,30,29)(19,24)(20,23)(22,36)(25,35)(28,37,34,32);(7,13)(8,17,18,35)(9,22)(10,23)(12,27,16,31)(14,32)(19,26,36,20)(21,37)(24,25)(28,29,34,33)

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, 6533);