C₅×#13721 #11936441

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#13721

Subgroups

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

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

normal subgroups only

This group has 180 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) = 640

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

Conjugacy classes & character table

350 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6709);