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C3⋊(C3⋊(C7⋊AGL(1,𝔽11))) #20294168

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

Aliases

unambiguous semidirect productC3⋊(C3⋊(C7⋊AGL(1,𝔽11)))
unambiguous semidirect productC7⋊(C3⋊(C3⋊AGL(1,𝔽11)))
unambiguous semidirect product(C11⋊C5)⋊(C3⋊D21)
unambiguous semidirect product(C3⋊D231)⋊C5

Splittings

Fitting, socle, derived, HallC32×C7×C11 ⋊ C2×C5
HallC32×C7×C11⋊C5 ⋊ C2
HallC3⋊D231 ⋊ C5
HallC32×C11⋊C5 ⋊ D7
HallC7×C11⋊C5 ⋊ C3⋊D3
HallC32×C11 ⋊ C5×D7
HallC7×C11 ⋊ C5×C3⋊D3
HallC32×C7 ⋊ AGL(1,𝔽11)
HallC11⋊C5 ⋊ C3⋊D21
SylowC11 ⋊ C5×C3⋊D21
SylowC32 ⋊ C7⋊AGL(1,𝔽11)
SylowC7 ⋊ C3⋊(C3⋊AGL(1,𝔽11))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC32×C7×C11C2×C5
LCSG ▷ C32×C7×C11
UCSC1

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽11)×C3⋊D21, D7×C3⋊(C3⋊AGL(1,𝔽11)), C3⋊D3×C7⋊AGL(1,𝔽11), …
maximal normal subgroup ofAGL(1,𝔽11)×C3⋊D21, D7×C3⋊(C3⋊AGL(1,𝔽11)), C3⋊D3×C7⋊AGL(1,𝔽11), …
maximal quotient ofC3⋊(C3⋊(C7⋊(Dic11⋊C5))), C2×C3⋊(C3⋊(C7⋊AGL(1,𝔽11))), C3⋊(C3⋊(C7⋊(D11⋊C25))), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6930, 55);