C₃×#20964 #10639571

order	768
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₃×#20964

Subgroups

	subgroup	G/H
center	C₃×C₄	C₂×#120
commutator	C₂³	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂×C₃	C₂×#386
prime radical	C₃×#3449	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂³×C₃×C₄ ◁ G

1700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#20964, #72253901
maximal subgroup of	C₃×#6437576, D₃×#20964, #72253901
maximal normal subgroup of	C₃×#6437576, D₃×#20964, #72253901

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13952);

C3×#20964 #10639571

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×#20964 #10639571