C₅×#13524 #11936244

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13523
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13524

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2092
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

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

normal subgroups only

This group has 288 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5)(6,7,10,16,35,18,39,51,13,20,11,17,12,19,36,52)(8,21,23,55,24,53,37,43,26,57,62,42,25,49,33,14)(9,27,29,47,40,50,38,15,32,45,30,46,31,54,34,48)(22,41,58,63,28,64,65,69,60,67,66,68,59,44,61,56);(6,8,12,24,13,26,35,25)(7,14,19,55,20,43,18,42)(9,28,31,60,32,59,40,22)(10,33,36,23,11,37,39,62)(15,44,47,41,48,64,46,67)(16,49,52,21,17,53,51,57)(27,63,54,69,45,68,50,56)(29,58,34,65,30,66,38,61);(6,9,13,32)(7,15,20,48)(8,22,26,60)(10,34,11,38)(12,40,35,31)(14,41,43,67)(16,50,17,54)(18,47,19,46)(21,56,57,69)(23,61,62,65)(24,59,25,28)(27,52,45,51)(29,39,30,36)(33,58,37,66)(42,64,55,44)(49,63,53,68)

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, 6512);