C₅×#13356 #11936076

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13353
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13356

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

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

normal subgroups only

This group has 676 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)(8,18,22,40)(9,23,26,42)(10,27,30,15)(11,31,33,16)(13,28,36,20)(14,32,39,25)(19,45,48,69)(21,51,53,44)(24,56,58,55)(29,41,63,60)(34,65,37,68)(35,50,64,61)(38,57,59,62)(43,52,66,47)(46,54,49,67);(6,8)(7,13)(9,19)(10,28)(11,21)(12,22)(14,37)(15,40)(16,35)(17,36)(18,27)(20,30)(23,54)(24,49)(25,47)(26,48)(29,61)(31,64)(32,52)(33,53)(34,39)(38,66)(41,51)(42,67)(43,59)(44,60)(45,56)(46,58)(50,63)(55,69)(57,68)(62,65);(6,9,12,26)(7,14,17,39)(8,19,22,48)(10,24,30,58)(11,32,33,25)(13,34,36,37)(15,38,27,59)(16,42,31,23)(18,43,40,66)(20,49,28,46)(21,52,53,47)(29,62,63,57)(35,54,64,67)(41,56,60,55)(44,69,51,45)(50,65,61,68)

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, 6344);