C₅₃×C₂₇⋊C₃ #16864293

order	4293
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₉×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

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

Subgroups

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

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 1431

A minimal faithful permutation representation — 2 generators on 80 points:

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

Conjugacy classes & character table

1749 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅₃×#6869, C₅₃×#6872, C₅₃×#6873, …
maximal normal subgroup of	C₅₃×#6869, C₅₃×#6872, C₅₃×#6873, …
maximal quotient of	C₅₃×#6864, C₅₃×C₃²⋊C₂₇, C₅₃×C₉⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4293, 6);

C53×C27⋊C3 #16864293