C₁₉×#6872 #17987258

order	4617
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃×C₉×C₁₉ · Schur multiplier (H₂) = C₃

Aliases

direct product

C₁₉×#6872

Subgroups

	subgroup	G/H
center	C₉×C₁₉	H(3,𝔽₃)
commutator	C₃²	C₃×C₉×C₁₉
Frattini, cosocle	C₃×C₉	C₃²×C₁₉
socle	C₃×C₁₉	C₃²⋊C₉
prime radical	C₁₉×H(3,𝔽₃)	C₉
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉×C₁₉ ◁ C₃×C₉×C₁₉ ◁ G

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 100
transitive degree	μ_t(G) = 1539

A minimal faithful permutation representation — 2 generators on 100 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,25,23,28,40,35,46,62,24,29,39,36,47,61,57,71,84,37,27,43,58,45,67,81,70,88)(22,30,48,33,50,73,55,75,92,34,44,66,56,69,87,80,90,98,32,26,38,54,42,60,79,65,83)(31,49,59,52,74,82,77,93,96,51,68,72,76,89,91,94,99,100,53,41,63,78,64,85,95,86,97);(20,22,31)(21,26,41)(23,33,52)(24,34,51)(25,38,59)(27,44,68)(28,42,64)(29,30,49)(32,53,37)(35,55,77)(36,56,76)(39,48,72)(40,60,82)(43,66,63)(45,69,89)(46,65,86)(47,50,74)(54,78,58)(57,80,94)(61,73,91)(62,83,96)(67,87,85)(70,90,99)(71,75,93)(79,95,81)(84,92,100)(88,98,97)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₁₉⋊C₃×#6872, #28354705, #28354706, …
maximal subgroup of	C₃×C₁₉×#6872, C₁₉×#10624702, C₁₉×#10624724, …
maximal normal subgroup of	C₃×C₁₉×#6872, C₁₉×#10624702, C₁₉×#10624724, …
maximal quotient of	C₁₉×H(3,𝔽₃)⋊C₂₇, C₁₉×3₋¹⁺²⋊C₂₇, C₁₉×#10624453, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4617, 19);

C19×#6872 #17987258