C₅×#9973 #11932693

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#9973

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	#120
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ G

632 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 138 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)(6,7)(8,9)(10,11,15,12)(13,16,14,17)(18,19,24,21)(20,27,30,33)(22,32,23,26)(25,29,31,28);(11,16)(12,17)(13,14)(18,20,24,30)(19,25,21,31)(22,29,23,28)(26,33,32,27);(7,9)(10,12,15,11)(13,17,14,16)(18,21,24,19)(20,27,30,33)(22,32,23,26)(25,28,31,29)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2961);