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ΓL(1,𝔽8)×#688 #13132049

order1680
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×#688
unambiguous semidirect product(C7×#688)⋊C3

Splittings

FittingC5×C7×Q8 ⋊ C2×C3
HallC7×#688 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ SD16
HallC5×C7 ⋊ C3×SD16
SylowC7 ⋊ C3×#688
SylowC5 ⋊ SD16×ΓL(1,𝔽8)

Subgroups

subgroupG/H
centerC2ΓL(1,𝔽8)×C22⋊D5
commutator, cosocleC4×C5×C7C22×C3
FrattiniC4C2×D5×ΓL(1,𝔽8)
FittingC5×C7×Q8C2×C3
socleC2×C5×C7C3×D4
prime radicalΓL(1,𝔽8)×D20C2
hypercenterQ8D5×ΓL(1,𝔽8)
hypocenterC5×C7C3×SD16
LCSG ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C4 ◁ Q8

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 280

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×SD16×ΓL(1,𝔽8), ΓL(1,𝔽8)×#4060, ΓL(1,𝔽8)×#4095, …
maximal normal subgroup ofD5×SD16×ΓL(1,𝔽8), ΓL(1,𝔽8)×#4060, ΓL(1,𝔽8)×#4095, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 69);