C₅×#13697 #11936417

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#13697

Subgroups

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

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

normal subgroups only

This group has 210 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) = 41
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(6,7)(8,9)(11,16)(12,17)(13,21)(14,24)(18,26)(19,35)(20,36)(22,38)(23,32)(25,37)(27,28)(29,34)(30,31)(33,41)(39,40);(10,11,17,28,13,19,30,36,24,34,38,41,37,39,23,26,15,18,32,40,25,33,22,29,14,20,31,35,21,27,12,16);(7,9)(11,18)(16,26)(19,33)(20,34)(27,39)(28,40)(29,36)(35,41)

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, 6685);