C₅×#13724 #11936444

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#13724

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

174 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 77
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 77 points:

(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,22,29)(16,24,34,54)(17,23,37,51)(18,26,40,47)(19,25,30,56)(20,43,45,72)(21,46,44,28)(27,59,60,38)(31,52,61,76)(32,65,67,77)(33,68,66,53)(35,48,69,73)(36,70,63,50)(39,71,41,58)(42,57,62,55)(49,74,75,64);(7,10)(11,13)(14,16,30,61,20,32,39,36,44,66,42,64,59,69,18,17,22,34,19,31,45,67,41,63,21,33,62,74,38,35,40,37)(15,23,47,73,27,49,55,53,46,70,58,65,72,76,25,24,29,51,26,48,60,75,57,68,28,50,71,77,43,52,56,54);(6,8,9,12)(7,11,10,13)(14,17,18,35,38,64,42,33,21,36,39,67,45,61,30,34,22,37,40,69,59,74,62,66,44,63,41,32,20,31,19,16)(15,24,25,52,43,65,58,50,28,53,55,75,60,73,47,51,29,54,56,76,72,77,71,70,46,68,57,49,27,48,26,23)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6712);