C₃×#89424 #13055086

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂×C₄

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₃×#89424

Splittings

Hall	C₃×C₇ ⋊ #387
Sylow	C₇ ⋊ C₃×#387

Subgroups

	subgroup	G/H
center	C₂³×C₃	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₃×C₄
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₃×C₇×#95	C₂
socle, prime radical	C₂³×C₃×C₇	C₂³
cosocle	C₂³×C₇	C₂³×C₃
hypercenter	C₃×#95	D₇
hypocenter	C₇	C₃×#387
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#95

308 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 1344

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4375);

C3×#89424 #13055086