C₇×#4675 #13055711

order	1344
isoclinism	isoclinic class of #4656
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₇×#4675

Splittings

Fitting	C₃×C₇×#95 ⋊ C₂
Hall	C₃×C₇ ⋊ #397
Sylow	C₃ ⋊ C₇×#397

Subgroups

	subgroup	G/H
center	C₂³×C₇	C₂²×D₃
commutator, cosocle	C₂³×C₃	C₂³×C₇
Frattini	C₂³	C₂²×D₃×C₇
Fitting	C₃×C₇×#95	C₂
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂³×D₃×C₇	C₂²
hypercenter	C₇×#95	D₃
hypocenter	C₃	C₇×#397
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₇ ◁ C₇×#95

340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 672

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5000);

C7×#4675 #13055711