C₅×#21828 #11944548

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×#21828

Subgroups

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

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

normal subgroups only

This group has 990 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) = 320

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

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

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, 14816);