C₂×C₃×#19153444 #46829418

order	34596
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃×C₃₁

Aliases

direct product

C₂×C₃×#19153444

Splittings

Fitting, socle	C₂×C₃×C₃₁² ⋊ C₂×C₃
derived, Sylow	C₃₁² ⋊ C₂²×C₃²
Hall	C₃×#15668333 ⋊ C₂²
Hall	C₂×C₃₁⋊D₃₁ ⋊ C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#19153444
commutator, cosocle, hypocenter	C₃₁²	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₃₁²	C₂×C₃
LCS	G ▷ C₃₁²
UCS	C₁ ◁ C₂×C₃

220 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:

(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)(47,60,61,53,54,66)(48,59,55,56,52,49)(57,67,64,63,62,65);(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)(37,38,41,47,55,43,42,48,57,64,51,59,50,54,62,61,66,67,60,46,52,58,65,63,56,40,44,49,53,45,39)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34596, 178);

C2×C3×#19153444 #46829418