C₅×#13684 #11936404

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

Aliases

direct product

C₅×#13684

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂³×C₅
prime radical	C₅×#3293	C₂
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₅×C₄⋊Q₈ ◁ G

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

normal subgroups only

This group has 378 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)(6,7,13,12)(8,17,21,31)(9,22,24,26)(10,11,27,16)(14,25,20,19)(15,28,29,23)(18,30,34,36)(32,35,37,33);(6,8,13,21)(7,14,12,20)(9,18,24,34)(10,25,27,19)(11,17,16,31)(15,30,29,36)(22,35,26,33)(23,37,28,32);(6,9)(7,15)(8,18)(10,26)(11,28)(12,29)(13,24)(14,30)(16,23)(17,32)(19,35)(20,36)(21,34)(22,27)(25,33)(31,37)

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, 6672);