C₂×C₃⋊#19153443 #46829430

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

Aliases

direct product	C₂×C₃⋊#19153443
unambiguous semidirect product	(C₂×C₃)⋊#19153443

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⋊#19153443
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)(7,11,12,8,9,13)(10,19,20,14,15,27)(16,30,31,22,23,35)(17,29,25,24,21,18)(26,28,36,32,33,34)(38,42,43,39,40,44)(41,50,51,45,46,58)(47,61,62,53,54,66)(48,60,56,55,52,49)(57,59,67,63,64,65);(1,2)(3,4,5)(37,38,41,47,55,40,44,49,57,64,46,52,58,62,67,54,61,65,66,51,60,50,59,63,56,43,42,48,53,45,39);(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)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34596, 190);

C2×C3⋊#19153443 #46829430