C₅×#32873 #11955593

order	1280
nilpotency class	3
isoclinism	isoclinic class of #32873
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₅×#32873

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2840
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₅
prime radical	C₂³×C₅×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂⁴×C₅ ◁ G

854 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2,3,4,5)(6,7)(8,14)(9,17)(10,13)(11,16)(12,18)(15,20)(19,21);(6,8)(7,11,13,14)(9,15)(10,16)(12,20,17,19)(18,21);(6,9)(7,12,13,17)(8,15)(10,18)(11,19,14,20)(16,21);(15,21)(19,20)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 25861);