C₅×#12406 #11935126

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12406

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈×Q₈ ◁ G

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

normal subgroups only

This group has 282 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) = 1280

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

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5394);