C₁₉×#13647 #18019606

order	4864
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#13647

Subgroups

	subgroup	G/H
center	C₁₉×C₃₂	D₄
commutator	C₄	C₂²×C₁₆×C₁₉
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₉
socle	C₂×C₁₉	D₄×C₁₆
prime radical	C₁₉×#135	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₉×C₃₂ ◁ C₂×C₁₉×C₃₂ ◁ G

186 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 118 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)(20,21,25,31,26,32,47,61,24,33,44,62,45,64,75,82,27,34,48,63,49,65,77,83,46,30,57,59,76,60,51,78)(22,35,37,66,38,41,55,72,39,53,68,79,70,43,56,74,40,58,69,81,71,73,54,80,36,28,50,52,67,23,29,42);(20,22)(21,28)(23,32)(24,39)(25,37)(26,38)(27,40)(29,51)(30,58)(31,52)(33,35)(34,53)(36,46)(41,64)(42,61)(43,65)(44,68)(45,70)(47,55)(48,69)(49,71)(50,57)(54,77)(56,75)(59,81)(60,73)(62,66)(63,79)(67,76)(72,82)(74,83)(78,80);(20,23,25,42,26,35,47,66,24,41,44,72,45,53,75,79,27,43,48,74,49,58,77,81,46,73,57,80,76,28,51,52)(21,29,31,22,32,37,61,38,33,55,62,39,64,68,82,70,34,56,63,40,65,69,83,71,30,54,59,36,60,50,78,67)

Conjugacy classes & character table

2128 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 6635);