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ΓL(1,𝔽8)×D435 #32912522

order18270
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D435
unambiguous semidirect productC3⋊(ΓL(1,𝔽8)×D145)
unambiguous semidirect productC5⋊(ΓL(1,𝔽8)×D87)
unambiguous semidirect productC29⋊(ΓL(1,𝔽8)×D15)
unambiguous semidirect product(C7×D435)⋊C3

Splittings

Fitting, socle, derivedC3×C5×C7×C29 ⋊ C2×C3
HallC3×C5×ΓL(1,𝔽8)×C29 ⋊ C2
HallC3×ΓL(1,𝔽8)×C29 ⋊ D5
HallC5×C7×C29 ⋊ C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ D29
HallC7×C29 ⋊ C3×D15
HallC5×C29 ⋊ D3×ΓL(1,𝔽8)
HallC3×ΓL(1,𝔽8) ⋊ D145
HallC5×C7 ⋊ C3×D87
SylowC29 ⋊ ΓL(1,𝔽8)×D15
SylowC7 ⋊ C3×D435
SylowC5 ⋊ ΓL(1,𝔽8)×D87

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C5×C7×C29C2×C3
LCSG ▷ C3×C5×C7×C29
UCSC1

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3045

Conjugacy classes & character table

1095 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofAGL(1,𝔽7)×D435, (ΓL(1,𝔽8)⋊D29)⋊D15, ΓL(1,𝔽8)×D15×D29, …
maximal normal subgroup ofAGL(1,𝔽7)×D435, (ΓL(1,𝔽8)⋊D29)⋊D15, ΓL(1,𝔽8)×D15×D29, …
maximal quotient ofΓL(1,𝔽8)×Dic435, C2×ΓL(1,𝔽8)×D435

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18270, 60);