C₅×#10385 #11933105

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10385

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#393
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

396 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 90 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)(6,7,9,12)(8,14)(10,13,18,20)(11,19)(15,16)(17,21)(22,23)(24,27)(25,32,34,28)(26,29)(30,35,33,37)(31,36);(6,7)(8,15,17,11)(9,12)(10,13)(14,16,21,19)(18,20)(22,24,26,31)(23,27,29,36)(25,33)(28,37)(30,34)(32,35);(6,8)(7,11)(9,17)(10,19)(12,15)(13,21)(14,20)(16,18)(22,25,26,34)(23,28,29,32)(24,30,31,33)(27,35,36,37)

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, 3373);