D₃×#19129581 #45657285

order	32634
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×#19129581
unambiguous semidirect product	C₃⋊(C₂×#19129581)

Splittings

Fitting, derived	C₃×C₃₇×C₄₉ ⋊ C₂×C₃
Hall	C₃×#19129581 ⋊ C₂
Hall	C₃₇×C₄₉ ⋊ C₃×D₃
Sylow	C₄₉ ⋊ D₃×C₃₇⋊C₃
Sylow	C₃₇ ⋊ D₃×C₄₉⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₃₇×C₄₉	C₂×C₃
Frattini	C₇	D₃×#11715913
socle	C₃×C₇×C₃₇	C₂×ΓL(1,𝔽₈)
LCS	G ▷ C₃×C₃₇×C₄₉
UCS	C₁

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 5439

A minimal faithful permutation representation — 2 generators on 89 points:

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

Conjugacy classes & character table

1821 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32634, 65);

D3×#19129581 #45657285