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C2×ΓL(1,𝔽8)×#135 #13060733

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

Aliases

direct productC2×ΓL(1,𝔽8)×#135
unambiguous semidirect product(C2×C7×#135)⋊C3

Splittings

Fitting, HallC2×C7×#135 ⋊ C3
SylowC7 ⋊ C2×C3×#135

Subgroups

subgroupG/H
centerC2×C4D4×ΓL(1,𝔽8)
commutator, cosocleC4×C7C24×C3
FrattiniC4C24×ΓL(1,𝔽8)
FittingC2×C7×#135C3
socleC22×C7C2×C3×D4
hypercenterC2×#135ΓL(1,𝔽8)
hypocenterC7C2×C3×#135
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

532 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)(5,12)(6,10)(7,15)(8,11)(9,13)(14,17)(16,18)(19,20,21,23,22,24,25);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)

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, 10022);