C₅×#18727 #11941447

order	1280
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#18727

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

996 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 230 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)(8,13)(9,15)(10,14)(11,16)(12,17)(18,21)(19,20)(22,23)(24,32)(25,30)(26,29)(27,31)(28,34)(33,37)(35,36);(6,8)(7,12)(9,17)(10,19)(11,18)(13,14)(15,21)(16,20)(22,24,27,35)(23,28,31,37)(25,33,26,34)(29,36,30,32);(6,8)(7,13)(9,19)(10,17)(11,18)(12,14)(15,20)(16,21)(23,29)(25,26)(28,36)(30,31)(32,37)(33,34);(6,9,11,10)(7,14,16,15)(8,17,18,19)(12,13,20,21)(22,25,27,26)(23,29,31,30)(24,33,35,34)(28,36,37,32)

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, 11715);