C₅×#7447 #11930167

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7447

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1361
commutator	C₂×C₁₆	C₂×C₄×C₅
Frattini, cosocle	C₄×C₁₆	C₂²×C₅
prime radical	C₂×C₅×D₁₆	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

130 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 2 generators on 69 points:

(1,2,3,4,5)(6,7,10,17,35,20,37,18,14,21,11,19,13,40,12,38)(8,22,24,46,41,54,25,59,27,55,39,69,26,60,36,53)(9,28,30,50,64,48,66,61,34,62,31,16,33,47,32,49)(15,29,42,58,44,67,51,57,45,63,68,23,52,56,43,65);(6,8)(7,15)(9,29)(10,25)(11,36)(12,39)(13,41)(14,27)(16,46)(17,51)(18,42)(19,43)(20,52)(21,45)(22,48)(23,32)(24,37)(26,35)(28,54)(30,57)(31,65)(33,67)(34,63)(38,68)(40,44)(47,55)(49,53)(50,69)(56,64)(58,66)(59,61)(60,62)

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 435);