C₁₉×#12171 #18018130

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#12171

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1598
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₉
prime radical	C₁₉×#575	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×M₁₆×C₁₉ ◁ G

332 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 83
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 3 generators on 83 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,32)(23,37)(24,41)(25,47)(26,31)(27,34)(28,39)(29,46)(30,44)(33,60)(35,67)(36,59)(38,71)(40,53)(42,56)(43,76)(45,57)(48,54)(49,65)(50,66)(51,78)(52,79)(55,64)(58,62)(61,68)(63,81)(69,74)(70,73)(72,80)(75,83)(77,82);(20,22,26,36)(21,27,31,50)(23,33,39,62)(24,42,45,78)(25,48,44,35)(28,49,37,68)(29,54,55,67)(30,56,47,51)(32,57,59,41)(34,64,66,46)(38,72,43,77)(40,75,73,63)(52,81,69,83)(53,82,70,80)(58,71,60,76)(61,79,65,74);(20,23,26,39)(21,28)(22,33,36,62)(24,43,45,38)(25,40,44,73)(27,49)(29,52)(30,53)(31,37)(32,58)(34,65,66,61)(35,63,48,75)(41,76)(42,77,78,72)(46,79,64,74)(47,70)(50,68)(51,80)(54,81)(55,69)(56,82)(57,71)(59,60)(67,83)

Conjugacy classes & character table

475 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5159);