C₅×#7803 #11930523

order	1280
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7803

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

338 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 1280

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

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

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 791);