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C7×#5407 #13057127

order1344
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×#5407

Splittings

FittingC7×#131 ⋊ D3

Subgroups

subgroupG/H
center, socleC2×C7C4⋊S4
commutator, cosocle#41⋊C3C22×C7
FrattiniC4C2×C7×S4
FittingC7×#131D3
hypercenterC7×C8S4
hypocenterSL(2,𝔽3)C7×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C7 ◁ C4×C7 ◁ C7×C8

130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 39
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,3,4,5,6,7)(8,16,35,27,23,33,10,25,22,9,28,39,13,32,34,14,11,31,15,38,24,18,19,37)(12,17,29,36,21,30,20,26);(9,14)(10,15)(11,12)(16,26)(17,25)(18,27)(19,29)(20,28)(21,23)(22,24)(30,38)(31,39)(32,36)(33,37)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6416);