C₅×#8495 #11931215

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8491
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#8495

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

620 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 320

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

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

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, 1483);