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C3⋊(C7⋊(C7⋊AGL(1,𝔽11))) #30633524

order16170
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C7

Aliases

unambiguous semidirect productC3⋊(C7⋊(C7⋊AGL(1,𝔽11)))
unambiguous semidirect productC7⋊(C3⋊(C7⋊AGL(1,𝔽11)))
unambiguous semidirect product(C11⋊C5)⋊(C3⋊(C7⋊D7))
unambiguous semidirect product(C3⋊(C7⋊D77))⋊C5

Splittings

Fitting, socle, derived, HallC3×C72×C11 ⋊ C2×C5
HallC3×C72×C11⋊C5 ⋊ C2
HallC3⋊(C7⋊D77) ⋊ C5
HallC72×C11⋊C5 ⋊ D3
HallC72×C11 ⋊ C5×D3
HallC3×C11⋊C5 ⋊ C7⋊D7
HallC3×C72 ⋊ AGL(1,𝔽11)
HallC11⋊C5 ⋊ C3⋊(C7⋊D7)
SylowC72 ⋊ C3⋊AGL(1,𝔽11)
HallC3×C11 ⋊ C5×C7⋊D7
SylowC11 ⋊ C5×C3⋊(C7⋊D7)
SylowC3 ⋊ C7⋊(C7⋊AGL(1,𝔽11))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C72×C11C2×C5
LCSG ▷ C3×C72×C11
UCSC1

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 1617

A minimal faithful permutation representation — 3 generators on 28 points:

(2,6,7,5,10,3,4,11,8,9)(13,14)(16,17)(18,19)(20,21)(23,24)(25,26)(27,28);(1,2,5,10,4,7,11,6,9,8,3)(12,13,14)(22,23,25,27,28,26,24);(15,16,18,20,21,19,17)

Polycyclic presentation:

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

Conjugacy classes & character table

522 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×C3⋊(C7⋊AGL(1,𝔽11)), D21×C7⋊AGL(1,𝔽11), AGL(1,𝔽11)×C3⋊(C7⋊D7), …
maximal normal subgroup ofD7×C3⋊(C7⋊AGL(1,𝔽11)), D21×C7⋊AGL(1,𝔽11), AGL(1,𝔽11)×C3⋊(C7⋊D7), …
maximal quotient ofC3⋊(C7⋊(C7⋊(Dic11⋊C5))), C2×C3⋊(C7⋊(C7⋊AGL(1,𝔽11)))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16170, 69);