C₅×#21795 #11944515

order	1280
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

direct product

C₅×#21795

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂⁴	C₂⁴×C₅
prime radical	C₂×C₅×#534	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₅×C₂²⋊C₄ ◁ G

1538 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1538 conjugacy classes of subgroups — too many to draw. Only its 366 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 — 4 generators on 29 points:

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 14783);