C₂×D₃×#15667864 #46802950

order	34164
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₃×#15667864
unambiguous semidirect product	C₃⋊(C₂²×#15667864)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×#15667864)

Splittings

Fitting, socle	C₂×C₃×C₁₃×C₇₃ ⋊ C₂×C₃
derived	C₃×C₁₃×C₇₃ ⋊ C₂²×C₃
Hall	C₃×#15667864 ⋊ C₂²
Hall	C₁₃×C₇₃ ⋊ C₂×C₃×D₃
Sylow	C₇₃ ⋊ C₂×D₃×C₁₃⋊C₃
Sylow	C₁₃ ⋊ C₂×D₃×C₇₃⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃×#15667864
commutator, cosocle, hypocenter	C₃×C₁₃×C₇₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₃×C₇₃	C₂×C₃
LCS	G ▷ C₃×C₁₃×C₇₃
UCS	C₁ ◁ C₂

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 5694

A minimal faithful permutation representation — 2 generators on 91 points:

(4,5)(7,8,10)(9,12,15)(11,14,13)(16,18,17)(20,21,23)(22,25,30)(24,28,27)(26,32,40)(29,35,37)(31,38,34)(33,42,53)(36,45,50)(39,48,47)(41,51,44)(43,55,68)(46,58,65)(49,56,62)(52,63,60)(54,66,57)(59,71,80)(61,72,77)(64,76,75)(67,78,73)(69,81,70)(74,84,87)(79,86,85)(82,88,83)(89,91,90);(1,2)(3,4,5)(6,7,9,8,11,15,12,16,18,10,13,17,14)(19,20,22,26,33,43,56,35,21,24,29,36,46,59,72,45,25,31,39,49,61,74,84,58,32,41,52,64,68,80,87,71,42,54,67,79,81,53,65,77,55,69,82,89,88,66,40,50,62,75,85,90,91,78,51,30,37,47,60,73,83,86,63,38,23,27,34,44,57,70,76,48,28)

Conjugacy classes & character table

1914 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34164, 316);

C2×D3×#15667864 #46802950