C₂×C₃⋊#19151044 #46802945

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

Aliases

direct product	C₂×C₃⋊#19151044
unambiguous semidirect product	(C₂×C₃)⋊#19151044

Splittings

Fitting, socle	C₂×C₃×C₁₃×C₇₃ ⋊ C₂×C₃
derived	C₃×C₁₃×C₇₃ ⋊ C₂²×C₃
Hall	C₃×#15667865 ⋊ C₂²
Hall	C₁₃×C₇₃ ⋊ C₂×C₃×D₃
Sylow	C₇₃ ⋊ C₂×C₃⋊(D₁₃⋊C₃)
Sylow	C₁₃ ⋊ C₂×D₃×C₇₃⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊#19151044
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)(7,11,12,8,9,13)(10,17,16,14,15,18)(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)(49,61,63)(52,56,60)(54,66,57)(59,71,80)(62,74,78)(64,72,76)(67,79,73)(69,81,70)(75,84,88)(77,85,87)(82,89,83)(86,90,91);(1,2)(3,4,5)(6,7,10,9,13,16,15,17,18,12,11,14,8)(19,20,22,26,33,43,56,38,23,27,34,44,57,70,79,51,30,37,47,60,73,83,89,66,40,50,63,76,55,69,82,81,53,65,78,87,71,42,54,67,68,80,88,91,84,58,32,41,52,64,77,86,90,74,45,25,31,39,49,62,75,85,61,35,21,24,29,36,46,59,72,48,28)

Conjugacy classes & character table

1038 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34164, 311);

C2×C3⋊#19151044 #46802945