C₂×C₃⋊#19151039 #46802948

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

Aliases

direct product	C₂×C₃⋊#19151039
unambiguous semidirect product	(C₂×C₃)⋊#19151039

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×C₃⋊(D₇₃⋊C₃)

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊#19151039
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)(20,24,25,21,22,26)(23,32,33,27,28,42)(29,45,46,35,36,59)(30,44,40,38,34,31)(37,64,65,48,49,78)(39,62,63,51,47,56)(41,43,61,53,54,57)(50,66,75,52,67,81)(55,60,80,68,69,76)(58,85,83,72,73,71)(70,79,86,74,82,90)(77,91,89,87,88,84);(1,2)(3,4,5)(6,7,9,8,11,15,12,16,18,10,13,17,14)(19,20,23,29,37,50,51,38,22,26,31,41,55,70,72,54,28,34,42,56,71,84,87,69,36,47,57,59,75,86,91,82,49,66,73,76,78,65,80,85,67,64,79,88,90,81,46,61,62,45,60,77,89,83,63,33,44,32,43,58,74,68,53,40,25,24,30,39,52,48,35,27,21)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34164, 314);

C2×C3⋊#19151039 #46802948