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C3×#90353 #13060993

order1344
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C24

Aliases

direct productC3×#90353

Splittings

FittingC3×C7×#123 ⋊ C2
HallC3×C7 ⋊ #542
SylowC7 ⋊ C3×#542

Subgroups

subgroupG/H
centerC22×C3C23×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
FittingC3×C7×#123C2
socleC22×C3×C7C24
prime radicalC3×#6502C2
hypercenterC3×#123D7
hypocenterC7C3×#542
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×#123

432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 672

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10282);