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C3⋊(C17⋊(ΓL(1,𝔽8)⋊D7)) #28418416

order14994
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(C17⋊(ΓL(1,𝔽8)⋊D7))
unambiguous semidirect productC17⋊(C3⋊(ΓL(1,𝔽8)⋊D7))
unambiguous semidirect productΓL(1,𝔽8)⋊D357

Splittings

Fitting, socle, derivedC3×C72×C17 ⋊ C2×C3
HallC3×C7×C17×ΓL(1,𝔽8) ⋊ C2
HallC72×C17 ⋊ C3×D3
HallC3×C7×ΓL(1,𝔽8) ⋊ D17
SylowC72 ⋊ C3×D51
SylowC17 ⋊ C3⋊(ΓL(1,𝔽8)⋊D7)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C72×C17C2×C3
LCSG ▷ C3×C72×C17
UCSC1

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1071

Polycyclic presentation:

2,3^2,7^2,17|c^a=c^2,d^a=d^6,e^a=e^6,f^a=f^16,d^b=d^2

Conjugacy classes & character table

897 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD17×C3⋊(ΓL(1,𝔽8)⋊D7), D51×ΓL(1,𝔽8)⋊D7, D3×C17⋊(ΓL(1,𝔽8)⋊D7), …
maximal normal subgroup ofD17×C3⋊(ΓL(1,𝔽8)⋊D7), D51×ΓL(1,𝔽8)⋊D7, D3×C17⋊(ΓL(1,𝔽8)⋊D7), …
maximal quotient ofC3⋊(ΓL(1,𝔽8)⋊Dic119), C2×C3⋊(C17⋊(ΓL(1,𝔽8)⋊D7))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14994, 108);