C₃×C₁₉×#733 #17987293

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×#733

Subgroups

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

132 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);(4,5,8,7,11,17,15,10,21)(6,12,16,14,9,18,25,20,28)(13,23,26,24,22,27,30,19,29);(4,6,13,15,25,30,7,14,24)(5,9,19,10,12,22,11,20,23)(8,16,26,21,28,29,17,18,27)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	#28353243, #28353244, #28353248, …
maximal subgroup of	C₁₉×#10624478, C₁₉×#10624479, C₁₉×#10624481, …
maximal normal subgroup of	C₁₉×#10624478, C₁₉×#10624479, C₁₉×#10624481, …
maximal quotient of	C₃×C₁₉×#6858, C₃×C₁₉×#6860, C₃×C₁₉×#6861, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4617, 54);

C3×C19×#733 #17987293