C₅×#13679 #11936399

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

Aliases

direct product

C₅×#13679

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2196
commutator, Frattini, cosocle	#124	C₂³×C₅
prime radical	C₅×#3288	C₂
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#124 ◁ G

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

normal subgroups only

This group has 386 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)(6,7,11,17)(8,18,22,35)(9,15,25,14)(10,16,28,29)(12,23,32,19)(13,33,31,20)(21,30,26,34)(24,37,27,36);(6,8)(7,12)(9,23)(10,21)(11,22)(13,29)(14,35)(15,18)(16,31)(17,32)(19,25)(20,27)(24,33)(26,28)(30,36)(34,37);(6,8)(7,13)(9,20)(10,26)(11,22)(12,29)(14,35)(15,18)(16,32)(17,31)(19,24)(21,28)(23,27)(25,33)(30,37)(34,36)

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, 6667);