C₁₉×#13198 #18019157

order	4864
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#13198

Subgroups

	subgroup	G/H
center	C₈×C₁₉	#127
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

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

normal subgroups only

This group has 410 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)(20,21,24,29,25,30,40,51,26,31,41,52,42,53,66,77)(22,27,34,45,35,46,60,72,33,44,58,70,59,71,81,83)(23,28,38,49,39,50,55,68,37,48,62,74,65,76,80,78)(32,54,57,79,36,61,64,82,56,43,67,69,63,47,73,75);(20,22,24,34,25,35,40,60,26,33,41,58,42,59,66,81)(21,27,29,45,30,46,51,72,31,44,52,70,53,71,77,83)(23,36,62,73,39,56,80,57,37,63,38,64,65,32,55,67)(28,47,74,82,50,54,78,69,48,61,49,75,76,43,68,79);(20,23)(21,28)(22,32,33,56)(24,38)(25,39)(26,37)(27,43,44,54)(29,49)(30,50)(31,48)(34,57,58,67)(35,36,59,63)(40,55)(41,62)(42,65)(45,69,70,79)(46,47,71,61)(51,68)(52,74)(53,76)(60,64,81,73)(66,80)(72,75,83,82)(77,78)

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, 6186);