C₇×#4937 #13055971

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₇×#4937

Splittings

Hall	C₃×C₇ ⋊ #383
Sylow	C₃ ⋊ C₇×#383

Subgroups

	subgroup	G/H
center	C₂³×C₇	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₇
Frattini	C₂³	C₂²×D₃×C₇
Fitting	C₂×C₃×C₄²×C₇	C₂
socle, prime radical	C₂³×C₃×C₇	C₂³
cosocle	C₂³×C₃	C₂³×C₇
hypercenter	C₂×C₄²×C₇	D₃
hypocenter	C₃	C₇×#383
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇

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:

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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5260);

C7×#4937 #13055971