C₃×C₁₉×#731 #17987291

order	4617
nilpotency class	3
isoclinism	isoclinic class of AΣL(1,𝔽₂₇)
automorphism	?
homology	Abelianization (H₁) = C₃³×C₁₉ · Schur multiplier (H₂) = C₃³

Aliases

direct product

C₃×C₁₉×#731

Subgroups

	subgroup	G/H
center, socle	C₃²×C₁₉	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃³×C₁₉
prime radical	C₃×C₁₉×H(3,𝔽₃)	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₁₉ ◁ C₃³×C₁₉ ◁ G

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 1539

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2,3)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49);(4,5,8)(6,12,16)(7,11,17)(9,18,14)(10,21,15)(13,23,26)(19,29,30)(20,28,25)(22,27,24);(4,6,13,7,14,24,15,25,30)(5,9,19,11,20,23,10,12,22)(8,16,26,17,18,27,21,28,29)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	#731×#4294, #28354817, #28354818, …
maximal subgroup of	C₉×C₁₉×#731, C₁₉×#731⋊C₉, C₁₉×#10624731, …
maximal normal subgroup of	C₉×C₁₉×#731, C₁₉×#731⋊C₉, C₁₉×#10624731, …
maximal quotient of	C₃×C₁₉×#6857, C₃×C₁₉×#6858, C₃×C₁₉×#6859, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4617, 52);

C3×C19×#731 #17987291