C₇×#4657 #13055693

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

Aliases

direct product

C₇×#4657

Splittings

Hall	C₃×C₇ ⋊ #398
Sylow	C₃ ⋊ C₇×#398

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×#398
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₇ ◁ C₇×#95

300 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:

(1,2,3,4,5,6,7)(8,9,10)(12,14)(15,17,18,16)(20,24)(23,26);(11,12)(13,14)(16,17)(19,21,22,25)(20,23,24,26);(9,10)(15,16,18,17)(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, 4982);

C7×#4657 #13055693