C₂×C₃⋊#19153449 #46829420

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

Aliases

direct product	C₂×C₃⋊#19153449
unambiguous semidirect product	(C₂×C₃)⋊#19153449

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊#19153449
commutator, cosocle, hypocenter	C₃×#15668333	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₃₁²	D₃
LCS	G ▷ C₃×#15668333
UCS	C₁ ◁ C₂

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 67 points:

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

Conjugacy classes & character table

1062 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34596, 180);

C2×C3⋊#19153449 #46829420