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ΓL(1,𝔽8)×#585 #13062299

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

Aliases

direct productΓL(1,𝔽8)×#585

Splittings

Fitting, HallC7×#585 ⋊ C3
SylowC7 ⋊ C3×#585

Subgroups

subgroupG/H
centerC4C24×ΓL(1,𝔽8)
commutator, socle, cosocleC2×C7C25×C3
FrattiniC2C25×ΓL(1,𝔽8)
FittingC7×#585C3
hypercenter#585ΓL(1,𝔽8)
hypocenterC7C3×#585
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ #585

1620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 5 generators on 23 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#3585, ΓL(1,𝔽8)×#5950, ΓL(1,𝔽8)×H(7,𝔽2), …
maximal normal subgroup ofΓL(1,𝔽8)×#3585, ΓL(1,𝔽8)×#5950, ΓL(1,𝔽8)×H(7,𝔽2), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11588);