C₅×#13502 #11936222

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13502
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13502

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

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

normal subgroups only

This group has 276 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)(6,7,12,27,26,17,9,16,11,15,29,14,10,18,25,33)(8,19,36,37,22,31,21,34,23,13,28,32,20,35,24,30);(6,8)(7,13)(9,24)(10,20)(11,23)(12,28)(14,32)(15,19)(16,34)(17,35)(18,31)(21,25)(22,26)(27,37)(29,36)(30,33);(7,14,15,27)(8,20)(9,12)(10,26)(13,30,19,34)(16,18,33,17)(21,24)(22,23)(25,29)(31,37,35,32)

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, 6490);