C₅×#8633 #11931353

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#8633

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

320 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 130 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)(6,7,9,13,10,12,18,20)(8,14,15,21,16,11,17,19)(22,23)(24,33)(25,38,40,45)(26,41)(27,31)(28,32)(29,35)(30,39,46,42)(34,43,50,53)(36,44)(37,48)(47,49,52,51);(22,24,27,36,28,37,26,35)(23,29,41,48,32,44,31,33)(25,34,42,51,40,50,39,49)(30,43,38,52,46,53,45,47);(6,8,10,16)(7,11,12,14)(9,17,18,15)(13,21,20,19)(22,25)(23,30)(24,34)(26,39)(27,42)(28,40)(29,43)(31,45)(32,46)(33,47)(35,49)(36,51)(37,50)(38,41)(44,53)(48,52)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1621);