C₃×#10951 #10629558

order	768
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₃×#10951

Subgroups

	subgroup	G/H
center	C₃×C₄	#386
commutator	C₂×C₄	C₂×C₃×C₄²
Frattini, cosocle	C₂³×C₄	C₂³×C₃
socle	C₂×C₃	#1437
prime radical	C₂²×C₃×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂³×C₃×C₄ ◁ G

916 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(4,5)(6,10,15,25)(7,17)(8,20)(9,12)(11,19)(13,22)(14,28,30,31)(16,27,21,24)(18,32)(23,34,35,29)(26,33);(4,6)(5,10,12,25)(7,14)(8,21)(9,15)(11,23,26,35)(13,27,20,24)(16,22)(17,31,32,28)(18,30)(19,34)(29,33);(4,7,8,19,9,18,22,33)(5,11,13,17,12,26,20,32)(6,14,21,34,15,30,16,29)(10,23,27,31,25,35,24,28)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#10951, #72271756
maximal subgroup of	C₃×#375718, D₃×#10951, #72271756
maximal normal subgroup of	C₃×#375718, D₃×#10951, #72271756

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3939);

C3×#10951 #10629558

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×#10951 #10629558