C₅×#13734 #11936454

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13734

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

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

normal subgroups only

This group has 150 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) = 45
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(15,20)(16,22)(17,25)(18,28)(21,30)(23,39)(24,40)(26,42)(27,34)(29,41)(31,32)(33,36)(35,45)(37,38)(43,44);(6,8,9,13)(7,10,11,12)(15,21)(20,30)(23,35)(24,36)(31,43)(32,44)(33,40)(39,45);(14,15,22,32,17,23,37,40,28,36,42,45,41,43,27,30,19,21,34,44,29,35,26,33,18,24,38,39,25,31,16,20)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6722);