C₅×#13486 #11936206

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13373
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13486

Subgroups

	subgroup	G/H
center	C₄×C₅	#486
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2071
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

296 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 3 generators on 37 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6474);