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C5×C7×C11⋊AGL(1,𝔽7) #30633527

order16170
isoclinismisoclinic class of C11⋊AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C2×C3×C5×C7 · Schur multiplier (H2) = C1

Aliases

direct productC5×C7×C11⋊AGL(1,𝔽7)
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×C7×D11)
unambiguous semidirect product(C5×D77)⋊(C3×C7)

Splittings

Fitting, socle, HallC5×C72×C11 ⋊ C2×C3
derivedC7×C11 ⋊ C2×C3×C5×C7
HallC5×C7×C11×ΓL(1,𝔽8) ⋊ C2
HallC5×C7×D77 ⋊ C3
HallC7×C11×ΓL(1,𝔽8) ⋊ C2×C5
HallC7×D77 ⋊ C3×C5
HallC5×C7×ΓL(1,𝔽8) ⋊ D11
HallC72×C11 ⋊ C2×C3×C5
HallC5×C72 ⋊ C3×D11
HallC7×ΓL(1,𝔽8) ⋊ C5×D11
HallC5×C11 ⋊ C7×AGL(1,𝔽7)
SylowC72 ⋊ C3×C5×D11
SylowC11 ⋊ C5×C7×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC5×C7C11⋊AGL(1,𝔽7)
commutator, cosocle, hypocenterC7×C11C2×C3×C5×C7
FrattiniC1G
Fitting, socleC5×C72×C11C2×C3
LCSG ▷ C7×C11
UCSC1 ◁ C5×C7

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2310

Conjugacy classes & character table

1120 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC5×C7×D11×AGL(1,𝔽7), C5×D7×C11⋊AGL(1,𝔽7), C5×(ΓL(1,𝔽8)⋊D7)⋊D11, …
maximal normal subgroup ofC5×C7×D11×AGL(1,𝔽7), C5×D7×C11⋊AGL(1,𝔽7), C5×(ΓL(1,𝔽8)⋊D7)⋊D11, …
maximal quotient ofC5×C7×C11⋊(ΓL(1,𝔽8)⋊C4), C2×C5×C7×C11⋊AGL(1,𝔽7)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16170, 72);