C₁₉×#13612 #18019571

order	4864
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#13612

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#2127
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₄⋊Q₈	C₂³×C₁₉
prime radical	C₂×#41×C₁₉	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#536 ◁ G

304 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 86 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)(20,21,24,30,26,32,44,59)(22,33,35,57,37,50,64,76)(23,28,40,53,39,54,68,81)(25,31,46,58,47,29,43,56)(27,42,49,45,51,63,77,36)(34,60,61,79,38,65,66,82)(41,52,72,78,69,55,71,80)(48,73,74,67,75,70,83,62);(20,22,26,37)(21,27,32,51)(23,38,39,34)(24,42,44,63)(25,45,47,36)(28,48,54,75)(29,50,31,33)(30,57,59,76)(35,43,64,46)(40,70,68,73)(41,62,69,67)(49,58,77,56)(52,60,55,65)(53,79,81,82)(61,72,66,71)(74,78,83,80);(20,23)(21,28)(22,34)(24,40)(25,41)(26,39)(27,48)(29,55)(30,53)(31,52)(32,54)(33,60)(35,61)(36,62)(37,38)(42,73)(43,71)(44,68)(45,67)(46,72)(47,69)(49,74)(50,65)(51,75)(56,80)(57,79)(58,78)(59,81)(63,70)(64,66)(76,82)(77,83)

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, 6600);