C₂×C₃⋊#19151048 #46802943

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

Aliases

direct product	C₂×C₃⋊#19151048
unambiguous semidirect product	(C₂×C₃)⋊#19151048

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×C₃⋊(D₇₃⋊C₃)

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊#19151048
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)(20,24,25,21,22,26)(23,32,33,27,28,42)(29,45,46,35,36,59)(30,44,40,38,34,31)(37,64,65,48,49,77)(39,62,63,51,47,56)(41,43,61,53,54,57)(50,69,75,55,60,78)(52,79,80,67,66,74)(58,83,82,71,72,70)(68,88,89,81,76,90)(73,91,86,84,85,87);(1,2)(3,4,5)(6,7,10,9,13,16,15,17,18,12,11,14,8)(19,20,23,29,37,50,53,40,25,24,30,39,52,68,82,63,33,44,32,43,58,73,86,80,46,61,62,45,60,76,85,89,65,78,83,79,64,49,66,72,75,77,90,91,88,69,36,47,57,59,74,87,84,71,54,28,34,42,56,70,81,67,51,38,22,26,31,41,55,48,35,27,21)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34164, 309);

C2×C3⋊#19151048 #46802943