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C5×C11×ΓL(1,𝔽8)⋊D7 #30633537

order16170
isoclinismisoclinic class of ΓL(1,𝔽8)⋊D7
automorphism?
homologyAbelianization (H1) = C2×C3×C5×C11 · Schur multiplier (H2) = C1

Aliases

direct productC5×C11×ΓL(1,𝔽8)⋊D7
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×C11×D7)

Splittings

Fitting, socle, HallC5×C72×C11 ⋊ C2×C3
derived, SylowC72 ⋊ C2×C3×C5×C11
HallC5×C7×C11×ΓL(1,𝔽8) ⋊ C2
HallC5×C11×C7⋊D7 ⋊ C3
HallC7×C11×ΓL(1,𝔽8) ⋊ C2×C5
HallC11×C7⋊D7 ⋊ C3×C5
HallC5×C7×ΓL(1,𝔽8) ⋊ C2×C11
HallC72×C11 ⋊ C2×C3×C5
HallC5×C7⋊D7 ⋊ C3×C11
HallC5×C72 ⋊ C2×C3×C11
HallC7×ΓL(1,𝔽8) ⋊ C2×C5×C11
HallC7⋊D7 ⋊ C3×C5×C11

Subgroups

subgroupG/H
center, hypercenterC5×C11ΓL(1,𝔽8)⋊D7
commutator, cosocle, hypocenterC72C2×C3×C5×C11
FrattiniC1G
Fitting, socleC5×C72×C11C2×C3
LCSG ▷ C72
UCSC1 ◁ C5×C11

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1155

Conjugacy classes & character table

1210 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC5×D11×ΓL(1,𝔽8)⋊D7, C5×C11×D7×AGL(1,𝔽7), C5×(ΓL(1,𝔽8)⋊D7)⋊D11, …
maximal normal subgroup ofC5×D11×ΓL(1,𝔽8)⋊D7, C5×C11×D7×AGL(1,𝔽7), C5×(ΓL(1,𝔽8)⋊D7)⋊D11, …
maximal quotient ofC5×C11×ΓL(1,𝔽8)⋊Dic7, C2×C5×C11×ΓL(1,𝔽8)⋊D7

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16170, 82);