C₅×#9945 #11932665

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9938
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#9945

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
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

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

normal subgroups only

This group has 574 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)(6,7,11,14)(8,15,17,21)(9,13,10,18)(12,19,20,16)(22,23)(24,27)(25,28)(26,29)(30,33)(31,34)(32,36)(35,37);(6,8)(7,12)(9,16)(10,19)(11,17)(13,21)(14,20)(15,18)(22,24,26,31)(23,27,29,34)(25,30,32,37)(28,35,36,33);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,15,20,21)(22,25)(23,28)(24,30)(26,32)(27,33)(29,36)(31,37)(34,35)

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, 2933);