C₅×#13373 #11936093

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

Aliases

direct product

C₅×#13373

Subgroups

	subgroup	G/H
center	C₄×C₅	#486
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2023
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

364 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 110 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)(6,7)(8,17,19,34)(9,22)(10,15)(11,27)(12,16)(13,20,29,36)(14,26)(18,35,21,31)(23,33)(24,32,37,28)(25,30);(8,18,19,21)(9,23)(13,28,29,32)(14,30)(17,31,34,35)(20,37,36,24)(22,33)(25,26);(6,8)(7,13)(9,24)(10,21)(11,18)(12,19)(14,31)(15,32)(16,29)(17,33)(20,25)(22,34)(23,37)(26,36)(27,28)(30,35)

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, 6361);