C₂×D₃×#15668333 #46829427

order	34596
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃₁

Aliases

direct product	C₂×D₃×#15668333
unambiguous semidirect product	C₃⋊(C₂²×#15668333)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×#15668333)

Splittings

Fitting, socle	C₂×C₃×C₃₁² ⋊ C₂×C₃
derived	C₃×C₃₁² ⋊ C₂²×C₃
Hall	C₃×#15668333 ⋊ C₂²
Sylow	C₃₁² ⋊ C₂×C₃×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃×#15668333
commutator, cosocle, hypocenter	C₃×C₃₁²	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₃₁²	C₂×C₃
LCS	G ▷ C₃×C₃₁²
UCS	C₁ ◁ C₂

188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 558

A minimal faithful permutation representation — 2 generators on 67 points:

(4,5)(7,8,10)(9,12,17)(11,15,14)(13,19,26)(16,22,24)(18,20,21)(23,29,33)(25,30,31)(27,34,28)(32,35,36)(38,39,41)(40,43,48)(42,46,45)(44,50,57)(47,51,54)(49,55,52)(53,60,63)(56,62,61)(58,64,59)(65,67,66);(1,2)(3,4,5)(6,7,9,13,20,10,14,21,28,34,17,24,31,19,27,26,33,36,29,12,18,25,32,35,22,8,11,16,23,30,15)(37,38,40,44,51,39,42,47,53,60,43,49,56,57,63,50,58,65,64,48,54,61,66,67,55,41,45,52,59,62,46)

Conjugacy classes & character table

1938 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34596, 187);

C2×D3×#15668333 #46829427