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C3×#89455 #13055116

order1344
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C2

Aliases

direct productC3×#89455

Splittings

HallC3×C7 ⋊ #499
SylowC7 ⋊ C3×#499

Subgroups

subgroupG/H
centerC22×C3C2×D28
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×C3×D7
FittingC3×C4×C7×C8C2
socle, prime radicalC22×C3×C7C2×D4
hypercenterC3×C4×C8D7
hypocenterC7C3×#499
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×C8

172 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 1344

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4405);