C₅×#13352 #11936072

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13350
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13352

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

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

normal subgroups only

This group has 682 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)(6,7,12,17)(8,18,22,44)(9,23,26,41)(10,15,28,40)(11,29,31,16)(13,32,35,20)(14,30,38,25)(19,45,48,69)(21,51,53,43)(24,55,57,68)(27,58,60,39)(33,63,36,66)(34,50,61,62)(37,59,67,56)(42,52,64,47)(46,54,49,65);(6,8)(7,13)(9,19)(10,20)(11,21)(12,22)(14,36)(15,18)(16,34)(17,35)(23,54)(24,46)(25,47)(26,48)(27,50)(28,32)(29,61)(30,52)(31,53)(33,38)(37,42)(39,43)(40,44)(41,65)(45,55)(49,57)(51,58)(56,63)(59,66)(60,62)(64,67)(68,69);(6,9,12,26)(7,14,17,38)(8,19,22,48)(10,24,28,57)(11,30,31,25)(13,33,35,36)(15,37,40,67)(16,41,29,23)(18,42,44,64)(20,49,32,46)(21,52,53,47)(27,59,60,56)(34,54,61,65)(39,68,58,55)(43,69,51,45)(50,63,62,66)

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, 6340);