C₁₉×C₈₁⋊C₃ #17987263

order	4617
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₁₉×C₂₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×C₈₁⋊C₃
unambiguous semidirect product	C₈₁⋊(C₃×C₁₉)

Subgroups

	subgroup	G/H
center	C₁₉×C₂₇	C₃²
commutator	C₃	C₃×C₁₉×C₂₇
Frattini, cosocle	C₂₇	C₃²×C₁₉
socle	C₃×C₁₉	C₃×C₂₇
prime radical	C₃²×C₁₉	C₂₇
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₁₉×C₂₇ ◁ G

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

Conjugacy classes & character table

1881 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#28353348, #28353349, #28353353, …
maximal subgroup of	C₁₉×#10624500, C₁₉×#10624502, C₁₉×#10624503, …
maximal normal subgroup of	C₁₉×#10624500, C₁₉×#10624502, C₁₉×#10624503, …
maximal quotient of	C₁₉×#10624499, C₁₉×C₃²⋊C₈₁, C₁₉×C₉⋊C₈₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4617, 24);

C19×C81⋊C3 #17987263