C₅×#10614 #11933334

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

Aliases

direct product

C₅×#10614

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

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, 3602);