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C2×ΓL(1,𝔽8)×#137 #13060735

order1344
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC2×ΓL(1,𝔽8)×#137
unambiguous semidirect product(C2×C7×#137)⋊C3

Splittings

Fitting, HallC2×C7×#137 ⋊ C3
SylowC7 ⋊ C2×C3×#137

Subgroups

subgroupG/H
centerC22C2×D4×ΓL(1,𝔽8)
commutator, cosocleC4×C7C24×C3
FrattiniC4C24×ΓL(1,𝔽8)
FittingC2×C7×#137C3
socleC22×C7C2×C3×D4
prime radicalC2×#41×ΓL(1,𝔽8)C2
hypercenterC2×#137ΓL(1,𝔽8)
hypocenterC7C2×C3×#137
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 4 generators on 25 points:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10024);