D₃×#19129580 #45657286

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

Aliases

direct product	D₃×#19129580
unambiguous semidirect product	C₃⋊(C₂×#19129580)

Splittings

Fitting, derived	C₃×C₃₇×C₄₉ ⋊ C₂×C₃
Hall	C₃×#19129580 ⋊ 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×#11715912
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)(42,44,48)(43,47,55)(45,51,63)(46,53,60)(49,59,57)(50,62,71)(52,66,84)(54,70,58)(56,73,72)(61,78,77)(64,80,74)(65,83,69)(67,85,87)(68,81,75)(76,79,86)(82,89,88);(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)(41,63,87,61,58,77,74,47,83,44,80,67,82,76,55,42,53,89,51,75,88,73,54,70,49,48,84,66,43,71,45,68,62,64,59,56,85,50,86,60,57,78,72,46,69,52,65,79,81)

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, 66);

D3×#19129580 #45657286