C₅×#10613 #11933333

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10611
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#10613

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

276 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 640

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

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

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, 3601);