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C5×C3⋊(ΓL(1,𝔽8)⋊D23) #28408043

order14490
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×C3⋊(ΓL(1,𝔽8)⋊D23)
unambiguous semidirect productC3⋊(C5×ΓL(1,𝔽8)⋊D23)
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×D69)
unambiguous semidirect productC23⋊(C5×C3⋊AGL(1,𝔽7))
unambiguous semidirect productD483⋊(C3×C5)
unambiguous semidirect product(C5×D483)⋊C3

Splittings

Fitting, socleC3×C5×C7×C23 ⋊ C2×C3
derivedC3×C7×C23 ⋊ C2×C3×C5
HallC3×C5×ΓL(1,𝔽8)×C23 ⋊ C2
HallC3×ΓL(1,𝔽8)×C23 ⋊ C2×C5
HallC5×C7×C23 ⋊ C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ D23
HallC7×C23 ⋊ C3×C5×D3
HallC5×C23 ⋊ C3⋊AGL(1,𝔽7)
HallC3×ΓL(1,𝔽8) ⋊ C5×D23
HallC5×C7 ⋊ C3×D69
SylowC23 ⋊ C5×C3⋊AGL(1,𝔽7)
SylowC7 ⋊ C3×C5×D69

Subgroups

subgroupG/H
center, hypercenterC5C3⋊(ΓL(1,𝔽8)⋊D23)
commutator, cosocle, hypocenterC3×C7×C23C2×C3×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C23C2×C3
LCSG ▷ C3×C7×C23
UCSC1 ◁ C5

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2415

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC5×D23×C3⋊AGL(1,𝔽7), C5×AGL(1,𝔽7)×D69, C5×D3×ΓL(1,𝔽8)⋊D23, …
maximal normal subgroup ofC5×D23×C3⋊AGL(1,𝔽7), C5×AGL(1,𝔽7)×D69, C5×D3×ΓL(1,𝔽8)⋊D23, …
maximal quotient ofC5×C3⋊(ΓL(1,𝔽8)⋊Dic23), C2×C5×C3⋊(ΓL(1,𝔽8)⋊D23)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14490, 34);