C₅×#8743 #11931463

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8738
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#8743

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#381
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

450 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 118 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)(6,7,9,11,8,10,12,13)(14,15)(16,23,25,19)(17,22)(18,21)(20,24,27,29)(26,28);(7,10)(11,13)(14,16)(15,19)(17,24)(18,25)(20,22)(21,23)(26,29)(27,28);(7,10)(9,12)(14,16)(15,20,21,27)(17,24)(18,25)(19,22,23,28)(26,29)

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, 1731);