C₅×#10078 #11932798

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#10078

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

780 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 25
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2,3,4,5)(11,15)(12,13)(16,17)(19,22)(20,24)(23,25);(6,7)(8,9)(10,11)(12,17)(13,15)(14,16)(18,19)(20,25)(21,23)(22,24);(7,9)(10,11)(12,15)(13,17)(14,16)(20,24)(22,25)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3066);