C₅×#13681 #11936401

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

Aliases

direct product

C₅×#13681

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

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

normal subgroups only

This group has 398 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) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2,3,4,5)(7,12)(8,15)(9,18)(13,21)(14,16)(17,20);(6,7)(8,16)(9,17)(10,13)(11,14)(12,15)(18,21)(19,20);(6,7)(8,17)(9,16)(10,20)(11,14)(12,18)(13,19)(15,21)

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, 6669);