C₅×#13444 #11936164

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13444

Subgroups

	subgroup	G/H
center	C₄×C₅	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2054
prime radical	C₅×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

392 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 106 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)(8,9)(10,23)(11,31)(12,33)(13,18)(14,19)(15,25)(16,30)(17,29)(20,35)(21,37)(22,27)(24,28)(26,34)(32,36);(6,8)(7,14)(9,23)(10,28)(11,15)(12,34)(13,22)(16,26)(17,24)(18,35)(19,30)(20,32)(21,29)(25,33)(27,31)(36,37);(6,9,10,24,29,37,32,35,13,27,11,25,12,26,30,14)(7,15,16,22,33,20,31,21,18,28,36,8,17,19,23,34)

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, 6432);