C₅×#13706 #11936426

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13706

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

164 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 78 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)(16,24)(18,29)(19,31)(21,35)(22,37)(23,28)(25,42)(26,38)(27,41)(30,33)(32,39)(34,36)(40,45)(43,44);(6,8,9,13)(7,10,11,12)(14,15,17,20)(16,21,25,34)(18,22,27,32)(19,23,28,31)(24,36,42,35)(26,33,40,44)(29,39,41,37)(30,38,43,45);(14,16,17,25)(15,21,20,34)(18,30,27,43)(19,33,28,44)(22,38,32,45)(23,40,31,26)(24,41,42,29)(35,39,36,37)

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6694);