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C3⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽11)) #20294154

order6930
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽11))
unambiguous semidirect productΓL(1,𝔽8)⋊(C3⋊AGL(1,𝔽11))
unambiguous semidirect product(C11⋊C5)⋊(C3⋊AGL(1,𝔽7))
unambiguous semidirect product(C3⋊(C11⋊AGL(1,𝔽7)))⋊C5
unambiguous semidirect product(C3⋊(C7⋊AGL(1,𝔽11)))⋊C3

Splittings

Fitting, socle, derivedC3×C7×C11 ⋊ C2×C3×C5
HallC3×ΓL(1,𝔽8)×C11⋊C5 ⋊ C2
HallC3⋊(C11⋊AGL(1,𝔽7)) ⋊ C5
HallC3×C11×ΓL(1,𝔽8) ⋊ C2×C5
HallC7×C11⋊C5 ⋊ C3×D3
HallC7×C11 ⋊ C3×C5×D3
HallC3×ΓL(1,𝔽8) ⋊ AGL(1,𝔽11)
HallC11⋊C5 ⋊ C3⋊AGL(1,𝔽7)
SylowC11 ⋊ C5×C3⋊AGL(1,𝔽7)
SylowC7 ⋊ C3×C3⋊AGL(1,𝔽11)

Subgroups

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

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽11)×C3⋊AGL(1,𝔽7), AGL(1,𝔽7)×C3⋊AGL(1,𝔽11), D3×ΓL(1,𝔽8)⋊AGL(1,𝔽11), …
maximal normal subgroup ofAGL(1,𝔽11)×C3⋊AGL(1,𝔽7), AGL(1,𝔽7)×C3⋊AGL(1,𝔽11), D3×ΓL(1,𝔽8)⋊AGL(1,𝔽11), …
maximal quotient ofC3⋊(ΓL(1,𝔽8)⋊(Dic11⋊C5)), C2×C3⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽11)), C3⋊(ΓL(1,𝔽8)⋊(D11⋊C25)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6930, 41);