C₅×#10222 #11932942

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10222

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

484 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 114 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)(6,7,11,15)(8,16,19,20)(9,12,10,14)(13,17,21,18)(24,31)(25,26)(28,32)(29,35)(33,34)(36,37);(7,12)(9,10)(13,20)(14,15)(16,21)(17,18)(22,23)(24,32)(25,29)(26,35)(27,30)(28,34)(31,36)(33,37);(6,8,11,19)(7,13,15,21)(9,18,10,17)(12,20,14,16)(22,24)(23,28)(25,34)(26,31)(27,33)(29,37)(30,36)(32,35)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3210);