C₅×#13676 #11936396

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13673
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13676

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2196
commutator, Frattini, cosocle	#127	C₂³×C₅
prime radical	C₅×#3291	C₂
LCS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#127 ◁ G

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

normal subgroups only

This group has 394 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)(6,7,11,15)(8,12,18,26)(9,13,21,29)(10,22,23,14)(16,30,33,36)(17,25,34,35)(19,27,20,28)(24,32,37,31);(6,8,11,18)(7,12,15,26)(9,16,21,33)(10,17,23,34)(13,24,29,37)(14,25,22,35)(19,31,20,32)(27,30,28,36);(6,9)(7,13)(8,16)(10,19)(11,21)(12,24)(14,27)(15,29)(17,31)(18,33)(20,23)(22,28)(25,30)(26,37)(32,34)(35,36)

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, 6664);