C₅×#13647 #11936367

order	1280
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13647

Subgroups

	subgroup	G/H
center	C₅×C₃₂	D₄
commutator	C₄	C₂²×C₅×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₅
socle	C₂×C₅	D₄×C₁₆
prime radical	C₅×#135	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₃₂ ◁ C₂×C₅×C₃₂ ◁ G

186 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 118 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)(6,7,11,17,12,18,33,47,10,19,30,48,31,50,61,68,13,20,34,49,35,51,63,69,32,16,43,45,62,46,37,64)(8,21,23,52,24,27,41,58,25,39,54,65,56,29,42,60,26,44,55,67,57,59,40,66,22,14,36,38,53,9,15,28);(6,8)(7,14)(9,18)(10,25)(11,23)(12,24)(13,26)(15,37)(16,44)(17,38)(19,21)(20,39)(22,32)(27,50)(28,47)(29,51)(30,54)(31,56)(33,41)(34,55)(35,57)(36,43)(40,63)(42,61)(45,67)(46,59)(48,52)(49,65)(53,62)(58,68)(60,69)(64,66);(6,9,11,28,12,21,33,52,10,27,30,58,31,39,61,65,13,29,34,60,35,44,63,67,32,59,43,66,62,14,37,38)(7,15,17,8,18,23,47,24,19,41,48,25,50,54,68,56,20,42,49,26,51,55,69,57,16,40,45,22,46,36,64,53)

Conjugacy classes & character table

560 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6635);