C₅₃×H(3,𝔽₃)⋊C₉ #27166205

order	12879
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃×C₉×C₅₃ · Schur multiplier (H₂) = C₃²

Aliases

direct product	C₅₃×H(3,𝔽₃)⋊C₉
unambiguous semidirect product	H(3,𝔽₃)⋊(C₉×C₅₃)
unambiguous semidirect product	(H(3,𝔽₃)×C₅₃)⋊C₉

Subgroups

	subgroup	G/H
center, socle	C₃²×C₅₃	H(3,𝔽₃)
commutator	C₃²	C₃×C₉×C₅₃
Frattini, cosocle	C₃³	C₃²×C₅₃
prime radical	C₃×H(3,𝔽₃)×C₅₃	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃³×C₅₃ ◁ G

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 4293

A minimal faithful permutation representation — 2 generators on 71 points:

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

Conjugacy classes & character table

2703 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₅₃×H(3,𝔽₃)⋊C₉, C₅₃×#10624697, C₅₃×#10624698, …
maximal normal subgroup of	C₃×C₅₃×H(3,𝔽₃)⋊C₉, C₅₃×#10624697, C₅₃×#10624698, …
maximal quotient of	C₅₃×H(3,𝔽₃)⋊C₂₇, C₅₃×#10624449, C₅₃×#10624450, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12879, 17);

C53×H(3,𝔽3)⋊C9 #27166205