C₅×#13362 #11936082

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13360
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13362

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₅
prime radical	C₅×#585	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

670 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5)(6,7,13,18)(8,19,23,45)(9,24,29,49)(10,30,34,16)(11,35,12,17)(14,31,41,21)(15,28,44,27)(20,52,54,68)(22,57,36,51)(25,50,58,69)(26,61,32,59)(33,47,37,48)(38,62,67,60)(39,55,40,56)(42,53,66,65)(43,63,46,64);(6,8)(7,14)(9,25)(10,31)(11,22)(12,36)(13,23)(15,42)(16,45)(17,39)(18,41)(19,30)(20,28)(21,34)(24,38)(26,62)(27,54)(29,58)(32,60)(33,56)(35,40)(37,55)(43,52)(44,66)(46,68)(47,57)(48,51)(49,67)(50,61)(53,63)(59,69)(64,65);(6,9,13,29)(7,15,18,44)(8,20,23,54)(10,32,34,26)(11,28,12,27)(14,38,41,67)(16,46,30,43)(17,49,35,24)(19,50,45,69)(21,53,31,65)(22,58,36,25)(33,63,37,64)(39,42,40,66)(47,59,48,61)(51,68,57,52)(55,62,56,60)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6350);