C₅×#12436 #11935156

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12436
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12436

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1667
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ G

234 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 74 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)(8,17)(9,13)(10,22)(15,33)(16,27)(18,19)(20,34)(21,35)(23,28)(24,29)(25,26)(30,31)(36,37);(6,7,13,26,10,15,23,27,11,16,28,33,22,25,9,12)(8,14,17,29,19,30,34,37,21,32,35,36,20,31,18,24);(6,8)(7,14)(9,18)(10,19)(11,21)(12,24)(13,17)(15,30)(16,32)(20,22)(23,34)(25,31)(26,29)(27,37)(28,35)(33,36)

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5424);