C₅×#13438 #11936158

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13438

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

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

normal subgroups only

This group has 406 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)(6,7,13,18)(8,19,24,44)(9,25,30,58)(10,31,12,16)(11,33,35,17)(14,23,39,21)(15,40,43,28)(20,49,52,69)(22,54,34,48)(26,59,50,65)(27,57,29,56)(32,45,36,47)(37,60,66,67)(38,53,46,55)(41,64,68,51)(42,61,63,62);(6,8,13,24)(7,14,18,39)(9,26,30,50)(10,23,12,21)(11,34,35,22)(15,41,43,68)(16,44,31,19)(17,46,33,38)(20,27,52,29)(25,49,58,69)(28,60,40,67)(32,53,36,55)(37,63,66,42)(45,48,47,54)(51,61,64,62)(56,65,57,59);(6,9,10,27,13,30,12,29)(7,15,16,42,18,43,31,63)(8,20,21,50,24,52,23,26)(11,28,32,61,35,40,36,62)(14,37,19,41,39,66,44,68)(17,25,45,56,33,58,47,57)(22,51,53,67,34,64,55,60)(38,65,48,49,46,59,54,69)

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, 6426);