C₅×#13330 #11936050

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13330

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

558 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 146 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)(8,9)(14,15,20,25)(16,26,17,29)(18,32,37,45)(19,38,35,24)(21,30,22,27)(23,41,44,36)(28,43,31,42)(33,40,34,39);(7,9)(10,11)(12,13)(14,16,20,17)(15,21,25,22)(18,33,37,34)(19,31,35,28)(23,42,44,43)(24,40,38,39)(26,45,29,32)(27,41,30,36);(11,13)(14,17,20,16)(15,22,25,21)(18,34,37,33)(19,28,35,31)(23,43,44,42)(24,39,38,40)(26,32,29,45)(27,36,30,41)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6318);