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ΓL(1,𝔽8)×D451 #34038828

order18942
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D451
unambiguous semidirect productC11⋊(ΓL(1,𝔽8)×D41)
unambiguous semidirect productC41⋊(ΓL(1,𝔽8)×D11)
unambiguous semidirect product(C7×D451)⋊C3

Splittings

Fitting, socle, derived, HallC7×C11×C41 ⋊ C2×C3
HallC11×ΓL(1,𝔽8)×C41 ⋊ C2
HallC7×D451 ⋊ C3
HallΓL(1,𝔽8)×C41 ⋊ D11
HallC11×C41 ⋊ C2×ΓL(1,𝔽8)
HallC7×C41 ⋊ C3×D11
HallC11×ΓL(1,𝔽8) ⋊ D41
HallC7×C11 ⋊ C3×D41
SylowC41 ⋊ ΓL(1,𝔽8)×D11
SylowC11 ⋊ ΓL(1,𝔽8)×D41
SylowC7 ⋊ C3×D451

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C11×C41C2×C3
LCSG ▷ C7×C11×C41
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3157

Conjugacy classes & character table

1135 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofAGL(1,𝔽7)×D451, (C11⋊AGL(1,𝔽7))⋊D41, ΓL(1,𝔽8)×D11×D41, …
maximal normal subgroup ofAGL(1,𝔽7)×D451, (C11⋊AGL(1,𝔽7))⋊D41, ΓL(1,𝔽8)×D11×D41, …
maximal quotient ofΓL(1,𝔽8)×Dic451, C2×ΓL(1,𝔽8)×D451

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18942, 20);