C₁₉×#8514 #18014473

order	4864
nilpotency class	3
isoclinism	isoclinic class of #8510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₉×#8514

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#379
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₁₉×#3568	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

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

normal subgroups only

This group has 648 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)(20,21,24,31,27,22,33,37)(23,39,42,75,44,65,61,74)(25,47,45,57,50,73,62,68)(26,32,46,29,51,38,63,35)(28,52,54,79,34,64,53,41)(30,58,60,70,36,69,72,49)(40,76,43,59,78,83,80,71)(48,55,81,56,82,66,77,67);(20,22)(21,27)(23,34)(24,29)(25,47)(26,38)(28,44)(30,49)(31,46)(32,51)(33,35)(36,70)(37,63)(39,48)(40,53)(41,59)(42,56)(43,66)(45,72)(50,73)(52,74)(54,78)(55,80)(57,69)(58,68)(60,62)(61,67)(64,75)(65,82)(71,79)(76,81)(77,83);(20,23)(21,28)(22,34)(24,42)(25,48)(26,43)(27,44)(29,56)(30,59)(31,54)(32,55)(33,61)(35,67)(36,71)(37,53)(38,66)(39,47)(40,63)(41,49)(45,81)(46,78)(50,82)(51,80)(52,58)(57,75)(60,83)(62,77)(64,69)(65,73)(68,74)(70,79)(72,76)

Conjugacy classes & character table

760 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 1502);