C₃×#10950 #10629557

order	768
nilpotency class	3
isoclinism	isoclinic class of #1909
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₃×#10950

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

904 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)(6,15,18,11)(7,19)(8,23)(9,14)(10,13)(12,21)(16,30,31,35)(17,26,24,27)(20,33)(22,28)(25,32,29,34);(4,6,9,17,10,18,8,24)(5,11,14,27,13,15,23,26)(7,16,22,32,20,31,21,34)(12,25,19,30,28,29,33,35);(4,7,8,21,10,20,9,22)(5,12,14,19,13,28,23,33)(6,16,24,34,18,31,17,32)(11,25,27,30,15,29,26,35)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#10950, #72271774
maximal subgroup of	C₃×#375718, D₃×#10950, #72271774
maximal normal subgroup of	C₃×#375718, D₃×#10950, #72271774

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3938);

C3×#10950 #10629557

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×#10950 #10629557