C₅×#13682 #11936402

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13681
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13682

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂³×C₅
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₅×C₄⋊Q₈ ◁ G

384 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6670);