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ΓL(1,𝔽8)×#713 #13132572

order1680
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productΓL(1,𝔽8)×#713

Splittings

FittingC5×C7×Q8 ⋊ C2×C3
HallC7×#713 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ #41
HallC5×C7 ⋊ C3×#41
SylowC7 ⋊ C3×#713
SylowC5 ⋊ #41×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, FrattiniC2C22×D5×ΓL(1,𝔽8)
commutator, socle, cosocleC2×C5×C7C23×C3
FittingC5×C7×Q8C2×C3
hypercenterQ8D5×ΓL(1,𝔽8)
hypocenterC5×C7C3×#41
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ Q8

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×#41×ΓL(1,𝔽8), ΓL(1,𝔽8)×#4060, #72218245, …
maximal normal subgroup ofD5×#41×ΓL(1,𝔽8), ΓL(1,𝔽8)×#4060, #72218245, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 592);