C₅×#9016 #11931736

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#9016

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	#120
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁵×C₅	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ G

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

normal subgroups only

This group has 836 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) = 33
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 33 points:

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2004);