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ΓL(1,𝔽8)×#683 #13132044

order1680
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productΓL(1,𝔽8)×#683
unambiguous semidirect product(C7×#683)⋊C3

Splittings

HallC7×#683 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C4⋊C4
HallC5×C7 ⋊ C3×C4⋊C4
SylowC7 ⋊ C3×#683
SylowC5 ⋊ C4⋊C4×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, FrattiniC22C2×D5×ΓL(1,𝔽8)
commutatorC2×C5×C7C2×C3×C4
FittingC2×C4×C5×C7C2×C3
socle, cosocleC22×C5×C7C22×C3
prime radicalC22×C5×ΓL(1,𝔽8)C22
hypercenterC2×C4D5×ΓL(1,𝔽8)
hypocenterC5×C7C3×C4⋊C4
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C22 ◁ C2×C4

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 560

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

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 64);