C₅×#13431 #11936151

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13429
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13431

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2044
commutator, Frattini, cosocle	C₂×#41	C₂³×C₅
prime radical	C₂×C₅×#41	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

402 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 86 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)(8,18,22,39)(9,23,27,41)(10,28,31,15)(11,16,33,42)(13,29,36,20)(14,32,38,26)(19,45,24,54)(21,44,50,68)(25,56,58,53)(30,52,59,40)(34,63,65,69)(35,61,66,48)(37,57,60,62)(43,49,67,47)(46,64,55,51);(6,8,12,22)(7,13,17,36)(9,24,27,19)(10,29,31,20)(11,21,33,50)(14,34,38,65)(15,39,28,18)(16,35,42,66)(23,51,41,64)(25,46,58,55)(26,47,32,49)(30,61,59,48)(37,67,60,43)(40,68,52,44)(45,56,54,53)(57,63,62,69);(6,9,10,25,12,27,31,58)(7,14,15,37,17,38,28,60)(8,19,20,46,22,24,29,55)(11,32,30,62,33,26,59,57)(13,34,18,43,36,65,39,67)(16,41,40,56,42,23,52,53)(21,49,48,63,50,47,61,69)(35,51,44,54,66,64,68,45)

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, 6419);