C₅×#13708 #11936428

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13708

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

158 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 74 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)(6,7,9,11)(8,12,13,10)(14,15,17,21)(16,26,27,55)(18,34,31,56)(19,39,32,64)(20,41,33,25)(22,46,43,28)(23,49,44,37)(24,52,45,36)(29,60,58,77)(30,61,42,57)(35,53,62,75)(38,71,63,67)(40,50,65,74)(47,72,73,69)(48,59,54,68)(51,70,66,76);(6,8,9,13)(7,10,11,12)(15,21)(22,43)(23,44)(24,45)(25,41)(26,55)(34,56)(39,64)(47,73)(48,54)(50,74)(51,66)(53,75)(57,61)(60,77)(67,71);(14,16,17,27)(15,22,21,43)(18,35,31,62)(19,40,32,65)(20,42,33,30)(23,50,44,74)(24,53,45,75)(25,54,41,48)(26,56,55,34)(28,49,46,37)(29,52,58,36)(38,72,63,69)(39,47,64,73)(51,77,66,60)(57,67,61,71)(59,70,68,76)

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6696);