C₂×D₃×#15668332 #46829429

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

Aliases

direct product	C₂×D₃×#15668332
unambiguous semidirect product	C₃⋊(C₂²×#15668332)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×#15668332)

Splittings

Fitting, socle	C₂×C₃×C₃₁² ⋊ C₂×C₃
derived	C₃×C₃₁² ⋊ C₂²×C₃
Hall	C₃×#15668332 ⋊ C₂²
Sylow	C₃₁² ⋊ C₂×C₃×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃×#15668332
commutator, cosocle, hypocenter	C₃×C₃₁²	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₃₁²	C₂×C₃
LCS	G ▷ C₃×C₃₁²
UCS	C₁ ◁ C₂

748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 5766

A minimal faithful permutation representation — 3 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)(47,53,55)(49,51,52)(54,60,64)(56,61,62)(58,65,59)(63,66,67);(1,2)(3,4,5)(37,38,40,44,51,41,45,52,59,65,48,55,62,50,58,57,64,67,60,43,49,56,63,66,53,39,42,47,54,61,46);(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)

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, 189);

C2×D3×#15668332 #46829429