C₁₉×#7381 #18013340

order	4864
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₉×#7381

Subgroups

	subgroup	G/H
center	C₁₆×C₁₉	D₈
commutator	C₈	C₂×C₁₆×C₁₉
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₉
socle	C₂×C₁₉	D₄⋊C₁₆
prime radical	C₁₉×#135	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₁₉ ◁ C₂×C₁₆×C₁₉ ◁ C₄×C₁₆×C₁₉ ◁ G

120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 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 — 2 generators on 83 points:

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

Conjugacy classes & character table

1672 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 369);