C₇×#4660 #13055695

order	1344
isoclinism	isoclinic class of #4656
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#4660

Splittings

Hall	C₃×C₇ ⋊ #400
Sylow	C₃ ⋊ C₇×#400

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, prime radical	C₂³×C₃×C₇	C₂³
hypercenter	C₇×#95	D₃
hypocenter	C₃	C₇×#400
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₇ ◁ C₇×#95

268 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)(11,12,13,14)(16,20)(19,22)(25,30)(27,29);(12,14)(15,17,18,21)(16,19,20,22)(23,25)(24,27)(26,30)(28,29);(9,10)(15,16)(17,19)(18,20)(21,22)(23,24,26,28)(25,29,30,27)

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, 4984);

C7×#4660 #13055695