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C23×ΓL(1,𝔽8)⋊D19 #32913487

order18354
automorphism?
homologyAbelianization (H1) = C2×C3×C23 · Schur multiplier (H2) = C1

Aliases

direct productC23×ΓL(1,𝔽8)⋊D19
unambiguous semidirect productΓL(1,𝔽8)⋊(C23×D19)

Splittings

Fitting, socle, HallC7×C19×C23 ⋊ C2×C3
derived, HallC7×C19 ⋊ C2×C3×C23
HallC19×ΓL(1,𝔽8)×C23 ⋊ C2
HallC23×D133 ⋊ C3
HallΓL(1,𝔽8)×C23 ⋊ D19
HallC19×C23 ⋊ AGL(1,𝔽7)
HallC19×ΓL(1,𝔽8) ⋊ C2×C23
HallD133 ⋊ C3×C23
HallC7×C23 ⋊ C3×D19
HallΓL(1,𝔽8) ⋊ C23×D19
SylowC19 ⋊ C23×AGL(1,𝔽7)
SylowC7 ⋊ C3×C23×D19

Subgroups

subgroupG/H
center, hypercenterC23ΓL(1,𝔽8)⋊D19
commutator, cosocle, hypocenterC7×C19C2×C3×C23
FrattiniC1G
Fitting, socleC7×C19×C23C2×C3
LCSG ▷ C7×C19
UCSC1 ◁ C23

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3059

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC23×D19×AGL(1,𝔽7), (ΓL(1,𝔽8)⋊D19)⋊D23, D23×ΓL(1,𝔽8)⋊D19, …
maximal normal subgroup ofC23×D19×AGL(1,𝔽7), (ΓL(1,𝔽8)⋊D19)⋊D23, D23×ΓL(1,𝔽8)⋊D19, …
maximal quotient ofC23×ΓL(1,𝔽8)⋊Dic19, C2×C23×ΓL(1,𝔽8)⋊D19

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18354, 42);