C₅×#10379 #11933099

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10379

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#393
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 360 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) = 160

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

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

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, 3367);