C₅×#21827 #11944547

order	1280
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅×#21827

Subgroups

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

1130 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1130 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 4 generators on 37 points:

(1,2,3,4,5)(6,7)(8,16)(9,20,23,13)(10,14)(11,15)(12,19)(17,31,32,28)(18,26)(21,35,36,25)(22,34,37,29)(24,30)(27,33);(6,8,10,19)(7,12,14,16)(9,21,22,17)(11,18,24,33)(13,28,29,25)(15,27,30,26)(20,31,34,35)(23,36,37,32);(6,9,10,22)(7,13,14,29)(8,17,19,21)(11,23,24,37)(12,25,16,28)(15,20,30,34)(18,32,33,36)(26,31,27,35);(7,14)(8,18)(12,26)(13,29)(15,30)(16,27)(17,32)(19,33)(20,34)(21,36)(25,31)(28,35)

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, 14815);