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ΓL(1,𝔽8)×H(3,𝔽4) #13060722

order1344
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C42

Aliases

direct productΓL(1,𝔽8)×H(3,𝔽4)

Splittings

Fitting, HallC7×H(3,𝔽4) ⋊ C3
SylowC7 ⋊ C3×H(3,𝔽4)

Subgroups

subgroupG/H
center, FrattiniC22C24×ΓL(1,𝔽8)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×H(3,𝔽4)C3
hypercenterH(3,𝔽4)ΓL(1,𝔽8)
hypocenterC7C3×H(3,𝔽4)
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ H(3,𝔽4)

520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#5464, ΓL(1,𝔽8)×#3532, AGL(1,𝔽7)×H(3,𝔽4), …
maximal normal subgroup ofΓL(1,𝔽8)×#5464, ΓL(1,𝔽8)×#3532, AGL(1,𝔽7)×H(3,𝔽4), …
maximal quotient ofΓL(1,𝔽8)×#2679

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10011);