C₁₉×#13199 #18019158

order	4864
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#13199

Subgroups

	subgroup	G/H
center	C₈×C₁₉	#124
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂×C₁₉	#1955
prime radical	C₁₉×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₉ ◁ C₂²×C₈×C₁₉ ◁ G

406 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 138 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)(20,21,25,31,26,32,47,61,27,33,48,62,24,30,35,50)(22,34,39,63,37,52,64,60,38,58,46,74,44,28,49,53)(23,29,42,56,41,55,70,79,43,57,73,82,45,59,65,76)(36,66,69,81,40,51,71,78,68,54,75,80,67,77,72,83);(20,22,35,64,24,44,48,39,27,38,47,49,26,37,25,46)(21,28,50,63,30,58,62,53,33,52,61,74,32,34,31,60)(23,40,70,69,45,36,42,72,43,67,65,75,41,68,73,71)(29,54,79,78,59,51,56,81,57,66,76,83,55,77,82,80);(20,23)(21,29)(22,36,38,68)(24,45)(25,42)(26,41)(27,43)(28,51,52,77)(30,59)(31,56)(32,55)(33,57)(34,54,58,66)(35,65)(37,40,44,67)(39,69,46,75)(47,70)(48,73)(49,72,64,71)(50,76)(53,78,60,83)(61,79)(62,82)(63,80,74,81)

Conjugacy classes & character table

988 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 6187);