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C2×ΓL(1,𝔽8)×2−1+4 #13062298

order1344
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC2×ΓL(1,𝔽8)×2−1+4

Splittings

Fitting, HallC2×C7×2−1+4 ⋊ C3
SylowC7 ⋊ C2×C3×2−1+4

Subgroups

subgroupG/H
centerC22C24×ΓL(1,𝔽8)
commutator, cosocleC2×C7C25×C3
FrattiniC2C25×ΓL(1,𝔽8)
FittingC2×C7×2−1+4C3
socleC22×C7C24×C3
hypercenterC2×2−1+4ΓL(1,𝔽8)
hypocenterC7C2×C3×2−1+4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×2−1+4

1588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 25 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11587);