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ΓL(1,𝔽8)×D143 #20238802

order6006
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D143
unambiguous semidirect productC11⋊(ΓL(1,𝔽8)×D13)

Splittings

Fitting, socle, derived, HallC7×C11×C13 ⋊ C2×C3
HallC11×C13×ΓL(1,𝔽8) ⋊ C2
HallC7×D143 ⋊ C3
HallC13×ΓL(1,𝔽8) ⋊ D11
HallC11×ΓL(1,𝔽8) ⋊ D13
HallC11×C13 ⋊ C2×ΓL(1,𝔽8)
HallC7×C13 ⋊ C3×D11
HallC7×C11 ⋊ C3×D13
SylowC13 ⋊ ΓL(1,𝔽8)×D11
SylowC11 ⋊ ΓL(1,𝔽8)×D13
SylowC7 ⋊ C3×D143

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC7×C11×C13C2×C3
LCSG ▷ C7×C11×C13
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

365 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×D143, (C11⋊AGL(1,𝔽7))⋊D13, ΓL(1,𝔽8)×D11×D13, …
maximal normal subgroup ofAGL(1,𝔽7)×D143, (C11⋊AGL(1,𝔽7))⋊D13, ΓL(1,𝔽8)×D11×D13, …
maximal quotient ofΓL(1,𝔽8)×Dic143, C2×ΓL(1,𝔽8)×D143, C7⋊C9×D143, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6006, 30);