C₅₃×C₉⋊C₂₇ #27166209

order	12879
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₂₇×C₅₃ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₅₃×C₉⋊C₂₇
unambiguous semidirect product	C₉⋊(C₂₇×C₅₃)
unambiguous semidirect product	(C₉×C₅₃)⋊C₂₇

Subgroups

	subgroup	G/H
center	C₃×C₉×C₅₃	C₃²
commutator	C₃	C₃×C₂₇×C₅₃
Frattini, cosocle	C₃×C₉	C₃²×C₅₃
socle, prime radical	C₃²×C₅₃	C₃×C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₅₃ ◁ G

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 12879

A minimal faithful permutation representation — 2 generators on 89 points:

(2,4,7)(5,9,8)(10,11,14,12,15,20,17,22,27,13,16,21,18,23,28,25,30,33,19,24,29,26,31,34,32,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,2,5,3,4,8,6,7,9)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅₃×#10624462, C₃×C₅₃×C₉⋊C₂₇, C₅₃×#10624706, …
maximal normal subgroup of	C₅₃×#10624462, C₃×C₅₃×C₉⋊C₂₇, C₅₃×#10624706, …
maximal quotient of	C₅₃×#10624462, C₅₃×#10624463, C₅₃×C₉⋊C₈₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12879, 21);

C53×C9⋊C27 #27166209