C₅×#13678 #11936398

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13677
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13678

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2196
commutator, Frattini, cosocle	#124	C₂³×C₅
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#124 ◁ G

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

normal subgroups only

This group has 400 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)(7,12)(9,20)(13,29)(14,28)(15,32)(16,21)(17,34)(18,24)(23,25)(27,30)(31,35)(33,37);(6,7)(8,13)(9,21)(10,15)(11,16)(12,20)(14,26)(17,35)(18,30)(19,31)(22,37)(23,32)(24,33)(25,28)(27,36)(29,34);(6,8,11,19)(7,13,16,31)(9,22,20,36)(10,24,26,18)(12,27,21,37)(14,30,15,33)(17,25,34,23)(28,29,32,35)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6666);