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C3⋊(C5⋊(C11⋊AGL(1,𝔽7))) #20294182

order6930
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(C5⋊(C11⋊AGL(1,𝔽7)))
unambiguous semidirect productC5⋊(C3⋊(C11⋊AGL(1,𝔽7)))
unambiguous semidirect productC11⋊(C3⋊(C5⋊AGL(1,𝔽7)))
unambiguous semidirect productΓL(1,𝔽8)⋊D165
unambiguous semidirect productD1155⋊C3

Splittings

Fitting, socle, derivedC3×C5×C7×C11 ⋊ C2×C3
HallC3×C5×C11×ΓL(1,𝔽8) ⋊ C2
HallC3×C11×ΓL(1,𝔽8) ⋊ D5
HallC5×C7×C11 ⋊ C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ D11
HallC7×C11 ⋊ C3×D15
HallC3×ΓL(1,𝔽8) ⋊ D55
HallC5×C11 ⋊ C3⋊AGL(1,𝔽7)
HallC5×C7 ⋊ C3×D33
SylowC11 ⋊ C3⋊(C5⋊AGL(1,𝔽7))
SylowC7 ⋊ C3×D165
SylowC5 ⋊ C3⋊(C11⋊AGL(1,𝔽7))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C7×C11C2×C3
LCSG ▷ C3×C5×C7×C11
UCSC1

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3^2,5,7,11|c^a=c^2,d^a=d^4,e^a=e^6,f^a=f^10,e^b=e^2

Conjugacy classes & character table

417 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD11×C3⋊(C5⋊AGL(1,𝔽7)), AGL(1,𝔽7)×D165, D15×C11⋊AGL(1,𝔽7), …
maximal normal subgroup ofD11×C3⋊(C5⋊AGL(1,𝔽7)), AGL(1,𝔽7)×D165, D15×C11⋊AGL(1,𝔽7), …
maximal quotient ofC3⋊(C11⋊(ΓL(1,𝔽8)⋊Dic5)), C2×C3⋊(C5⋊(C11⋊AGL(1,𝔽7))), C3⋊(C11⋊(ΓL(1,𝔽8)⋊D25)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6930, 69);