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C2×C3×C5×C7×ΓL(1,𝔽8) #17984396

order4410
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C32×C5×C7 · Schur multiplier (H2) = C3

Aliases

direct productC2×C3×C5×C7×ΓL(1,𝔽8)
unambiguous semidirect product(C2×C7)⋊(C32×C5×C7)
unambiguous semidirect product(C2×C3×C7)⋊(C3×C5×C7)
unambiguous semidirect product(C2×C5×C7)⋊(C32×C7)
unambiguous semidirect product(C2×C3×C5×C7)⋊(C3×C7)

Splittings

Fitting, socleC2×C3×C5×C72 ⋊ C3
derivedC7 ⋊ C2×C32×C5×C7
HallC2×C5×C72 ⋊ C32
HallC5×C72 ⋊ C2×C32
HallC2×C72 ⋊ C32×C5
SylowC72 ⋊ C2×C32×C5

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5×C7ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C2×C32×C5×C7
FrattiniC1G
Fitting, socleC2×C3×C5×C72C3
LCSG ▷ C7
UCSC1 ◁ C2×C3×C5×C7

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 630

Conjugacy classes & character table

1050 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×C3×C7×ΓL(1,𝔽8)×C25, C3×C5×C7×D5×ΓL(1,𝔽8), C2×C3×C52×C7×ΓL(1,𝔽8)
maximal normal subgroup ofC2×C3×C7×ΓL(1,𝔽8)×C25, C2×C3×C52×C7×ΓL(1,𝔽8)
maximal quotient ofC2×C3×C7×ΓL(1,𝔽8)×C25, C3×C5×C7×D5×ΓL(1,𝔽8), C2×C3×C52×C7×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4410, 117);