C₃×#10989 #10629596

order	768
nilpotency class	3
isoclinism	isoclinic class of #9103
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃×#10989

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#386
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂³×C₄	C₂³×C₃
prime radical	C₂×C₃×#535	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

1502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3)(4,5)(6,7)(8,9,12,16,13,15,11,20)(10,17,18,14,19,23,21,22);(8,10)(9,14)(11,21)(12,18)(13,19)(15,22)(16,23)(17,20);(5,7)(9,15)(11,12)(17,23)(18,21)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#10989, #72253536, #72254877, …
maximal subgroup of	C₃×#359509, D₃×#10989, #72253536, …
maximal normal subgroup of	C₃×#359509, D₃×#10989, #72253536, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3977);

C3×#10989 #10629596

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×#10989 #10629596