C₃×C₁₉×C₂₇⋊C₃ #17987288

order	4617
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₁₉ · Schur multiplier (H₂) = C₃²

Aliases

direct product	C₃×C₁₉×C₂₇⋊C₃
unambiguous semidirect product	C₂₇⋊(C₃²×C₁₉)

Subgroups

	subgroup	G/H
center	C₃×C₉×C₁₉	C₃²
commutator	C₃	C₃²×C₉×C₁₉
Frattini, cosocle	C₉	C₃³×C₁₉
socle	C₃²×C₁₉	C₃×C₉
prime radical	C₃³×C₁₉	C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₁₉ ◁ G

120 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);(4,5,8,6,10,16,12,19,24,7,11,15,13,20,23,21,27,29,14,9,17,22,18,25,28,26,30);(5,9,11)(8,15,17)(10,18,20)(16,23,25)(19,26,27)(24,29,30)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	#28353169, #28353173, #28353177, …
maximal subgroup of	C₁₉×#10624453, C₁₉×#10624454, C₁₉×#10624455, …
maximal normal subgroup of	C₁₉×#10624453, C₁₉×#10624454, C₁₉×#10624455, …
maximal quotient of	C₃×C₁₉×#6864, C₉×C₁₉×C₂₇⋊C₃, C₃×C₁₉×C₃²⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4617, 49);

C3×C19×C27⋊C3 #17987288