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ΓL(1,𝔽8)×#704 #13132402

order1680
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×#704
unambiguous semidirect product(C7×#704)⋊C3

Splittings

HallC7×#704 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ M16
HallC5×C7 ⋊ C3×M16
SylowC7 ⋊ C3×#704
SylowC5 ⋊ M16×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, FrattiniC2C2×AGL(1,𝔽5)×ΓL(1,𝔽8)
commutator, socleC2×C5×C7C2×C3×C4
FittingC22×C5×C7C3×C4
prime radicalC22×C5×ΓL(1,𝔽8)C4
cosocleC7×Dic5C22×C3
hypercenterC22AGL(1,𝔽5)×ΓL(1,𝔽8)
hypocenterC5×C7C3×M16
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C22

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×D5⋊M16, AGL(1,𝔽7)×#704, #72218296
maximal normal subgroup ofΓL(1,𝔽8)×D5⋊M16, AGL(1,𝔽7)×#704, #72218296

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 422);