C₅×#11653 #11934373

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11651
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11653

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂³×C₅	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

300 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 53
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7,9,12,10,13,16,19)(8,11,15,18,14,17,20,21)(22,23)(24,33,36,50)(25,29)(26,40)(27,32)(28,35,44,51)(30,38)(31,39)(34,49,37,46)(41,48,52,43)(42,45)(47,53);(6,8)(7,11)(9,15)(10,14)(12,18)(13,17)(16,20)(19,21)(22,24,27,36)(23,28,32,44)(25,34,39,37)(26,41,42,52)(29,43,31,48)(30,46,47,49)(33,40,50,45)(35,38,51,53);(8,14)(11,17)(15,20)(18,21)(22,25,27,39)(23,29,32,31)(24,34,36,37)(26,38,42,53)(28,43,44,48)(30,45,47,40)(33,49,50,46)(35,41,51,52)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4641);