C₅×#13435 #11936155

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13432
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13435

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

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

normal subgroups only

This group has 408 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)(8,18,22,40)(9,23,27,52)(10,28,31,15)(11,32,33,16)(13,29,36,20)(14,37,39,26)(19,44,24,53)(21,48,49,43)(25,56,58,51)(30,41,62,60)(34,64,66,69)(35,47,63,61)(38,57,59,67)(42,55,68,46)(45,65,54,50);(6,8,12,22)(7,13,17,36)(9,24,27,19)(10,29,31,20)(11,21,33,49)(14,34,39,66)(15,40,28,18)(16,35,32,63)(23,50,52,65)(25,45,58,54)(26,46,37,55)(30,61,62,47)(38,68,59,42)(41,48,60,43)(44,56,53,51)(57,64,67,69);(6,9,10,25,12,27,31,58)(7,14,15,38,17,39,28,59)(8,19,20,45,22,24,29,54)(11,26,30,57,33,37,62,67)(13,34,18,42,36,66,40,68)(16,23,41,51,32,52,60,56)(21,46,47,69,49,55,61,64)(35,65,43,44,63,50,48,53)

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, 6423);