C₅×#12460 #11935180

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12433
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#12460

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	#466
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

200 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 82 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)(6,7,10,13)(8,14,16,21)(9,17,19,12)(11,18,20,15)(23,27)(24,28)(25,31)(29,33)(30,37)(32,36)(34,35);(22,23,28,35,25,30,32,33,26,29,36,37,31,34,24,27);(6,8,10,16)(7,11,13,20)(9,18,19,15)(12,21,17,14)(23,29)(27,33)(30,34)(35,37)

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, 5448);