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ΓL(1,𝔽8)×D429 #31825068

order18018
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D429
unambiguous semidirect productC3⋊(ΓL(1,𝔽8)×D143)
unambiguous semidirect productC11⋊(ΓL(1,𝔽8)×D39)

Splittings

Fitting, socle, derivedC3×C7×C11×C13 ⋊ C2×C3
HallC3×C11×C13×ΓL(1,𝔽8) ⋊ C2
HallC7×C11×C13 ⋊ C3×D3
HallC3×C13×ΓL(1,𝔽8) ⋊ D11
HallC3×C11×ΓL(1,𝔽8) ⋊ D13
HallC11×C13 ⋊ D3×ΓL(1,𝔽8)
HallC7×C13 ⋊ C3×D33
HallC7×C11 ⋊ C3×D39
HallC3×ΓL(1,𝔽8) ⋊ D143
SylowC13 ⋊ ΓL(1,𝔽8)×D33
SylowC11 ⋊ ΓL(1,𝔽8)×D39
SylowC7 ⋊ C3×D429

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C7×C11×C13C2×C3
LCSG ▷ C3×C7×C11×C13
UCSC1

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3003

Conjugacy classes & character table

1080 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofAGL(1,𝔽7)×D429, (ΓL(1,𝔽8)⋊D13)⋊D33, (C11⋊AGL(1,𝔽7))⋊D39, …
maximal normal subgroup ofAGL(1,𝔽7)×D429, (ΓL(1,𝔽8)⋊D13)⋊D33, (C11⋊AGL(1,𝔽7))⋊D39, …
maximal quotient ofΓL(1,𝔽8)×Dic429, C2×ΓL(1,𝔽8)×D429

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18018, 113);