C₂×C₃⋊#19153445 #46829426

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

Aliases

direct product	C₂×C₃⋊#19153445
unambiguous semidirect product	(C₂×C₃)⋊#19153445

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	C₃⋊#19153445
commutator, cosocle, hypocenter	C₃×C₃₁²	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₃₁²	C₂×C₃
LCS	G ▷ C₃×C₃₁²
UCS	C₁ ◁ C₂

268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 372

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

(4,5)(7,11,12,8,9,13)(10,19,20,14,15,27)(16,30,31,22,23,35)(17,29,25,24,21,18)(26,28,36,32,33,34)(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)(6,7,10,16,24,9,13,18,26,33,15,21,27,31,36,23,30,34,35,20,29,19,28,32,25,12,11,17,22,14,8)(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

1008 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34596, 186);

C2×C3⋊#19153445 #46829426