C₅×#10095 #11932815

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10088
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10095

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

480 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 37
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3083);