C₃×C₅₃×C₂₇⋊C₃ #27166237

order	12879
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₅₃ · Schur multiplier (H₂) = C₃²

Aliases

direct product	C₃×C₅₃×C₂₇⋊C₃
unambiguous semidirect product	C₂₇⋊(C₃²×C₅₃)
unambiguous semidirect product	(C₂₇×C₅₃)⋊C₃²

Subgroups

	subgroup	G/H
center	C₃×C₉×C₅₃	C₃²
commutator	C₃	C₃²×C₉×C₅₃
Frattini, cosocle	C₉	C₃³×C₅₃
socle	C₃²×C₅₃	C₃×C₉
prime radical	C₃³×C₅₃	C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₅₃ ◁ G

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 4293

A minimal faithful permutation representation — 3 generators on 83 points:

(1,2,3)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(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

5247 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅₃×#10624453, C₅₃×#10624454, C₅₃×#10624455, …
maximal normal subgroup of	C₅₃×#10624453, C₅₃×#10624454, C₅₃×#10624455, …
maximal quotient of	C₃×C₅₃×#6864, C₉×C₅₃×C₂₇⋊C₃, C₃×C₅₃×C₃²⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12879, 49);

C3×C53×C27⋊C3 #27166237