C₅×#11759 #11934479

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#11759

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₅
prime radical	C₂⁵×C₅	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂⁴×C₅ ◁ C₂²×C₅×C₂²⋊C₄ ◁ G

586 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 29
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4747);