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C7×#395⋊C3 #13055050

order1344
automorphism?
homologyAbelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×#395⋊C3
unambiguous semidirect product#395⋊(C3×C7)

Splittings

Fitting, HallC7×#395 ⋊ C3
Sylow#395 ⋊ C3×C7

Subgroups

subgroupG/H
center, hypercenterC2×C7#120⋊C3
commutator, cosocle, hypocenterC22×Q8C2×C3×C7
FrattiniC23C2×C7×A4
FittingC7×#395C3
socleC23×C7C2×A4
prime radicalC7×C22⋊SL(2,𝔽3)C2
LCSG ▷ C22×Q8
UCSC1 ◁ C2×C7

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 168

A minimal faithful permutation representation — 2 generators on 31 points:

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

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4339);