C₃×#14727 #10633334

order	768
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₃×#14727

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₃
prime radical	C₂⁶×C₃	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

3350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#14727, #72248249, #72255551, …
maximal subgroup of	D₃×#14727, #72248249, C₃×#846041, …
maximal normal subgroup of	D₃×#14727, #72248249, C₃×#846041, …
maximal quotient of	C₃×#395781

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 7715);

C3×#14727 #10633334

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×#14727 #10633334