C₃×#24338 #10642945

order	768
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃×#24338

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₂²×C₃×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄×Q₈ ◁ G

1296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 384

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

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#24338, #72269974
maximal subgroup of	D₃×#24338, #72269974, C₃×#6453969
maximal normal subgroup of	D₃×#24338, #72269974, C₃×#6453969
maximal quotient of	C₃×#500940

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 17326);

C3×#24338 #10642945

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×#24338 #10642945