C₅×#8702 #11931422

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#8702

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁵×C₅	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂⁴×C₄×C₅ ◁ G

786 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 320

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15)(16,20,21,18)(17,19)(22,23,25,28,26,29,32,35)(24,27,31,34,30,33,36,37);(7,10)(11,13)(22,24)(23,27)(25,31)(26,30)(28,34)(29,33)(32,36)(35,37);(6,8,9,12)(7,11,10,13)(14,16)(15,18)(17,21)(19,20)(24,30)(27,33)(31,36)(34,37)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1690);