C₇×C₃⋊#401 #13055698

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

Aliases

direct product	C₇×C₃⋊#401
unambiguous semidirect product	C₃⋊(C₇×#401)

Splittings

Hall	C₃×C₇ ⋊ #401
Sylow	C₃ ⋊ C₇×#401

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×#401
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₇ ◁ C₇×#95

236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 1344

A minimal faithful permutation representation — 3 generators on 34 points:

(1,2,3,4,5,6,7)(8,9,10)(12,16)(15,18)(21,26)(23,25)(27,29,30,34)(28,31,32,33);(11,13,14,17)(12,15,16,18)(19,21)(20,23)(22,26)(24,25)(29,34)(31,33);(9,10)(11,12)(13,15)(14,16)(17,18)(19,20,22,24)(21,25,26,23)(27,28,30,32)(29,33,34,31)

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, 4987);

C7×C3⋊#401 #13055698