C₃×#89709 #13055369

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₃×#89709

Splittings

Hall	C₃×C₇ ⋊ #383
Sylow	C₇ ⋊ C₃×#383

Subgroups

	subgroup	G/H
center	C₂³×C₃	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₃×C₄
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₂×C₃×C₄²×C₇	C₂
socle, prime radical	C₂³×C₃×C₇	C₂³
cosocle	C₂³×C₇	C₂³×C₃
hypercenter	C₂×C₃×C₄²	D₇
hypocenter	C₇	C₃×#383
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄²

276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 1344

A minimal faithful permutation representation — 3 generators on 30 points:

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

Conjugacy classes & character table

372 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4658);

C3×#89709 #13055369