Finite Group Databasebeta

← Prev (#17984385) Next (#17984387) →

C7×ΓL(1,𝔽8)×D15 #17984386

order4410
automorphism?
homologyAbelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×ΓL(1,𝔽8)×D15
unambiguous semidirect productC3⋊(C7×D5×ΓL(1,𝔽8))
unambiguous semidirect productC5⋊(D3×C7×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D15)⋊(C3×C7)

Splittings

Fitting, socleC3×C5×C72 ⋊ C2×C3
derivedC3×C5×C7 ⋊ C2×C3×C7
HallC3×C5×C7×ΓL(1,𝔽8) ⋊ C2
HallC3×C7×ΓL(1,𝔽8) ⋊ D5
HallC5×C72 ⋊ C3×D3
SylowC72 ⋊ C3×D15
SylowC5 ⋊ D3×C7×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC7ΓL(1,𝔽8)×D15
commutator, cosocle, hypocenterC3×C5×C7C2×C3×C7
FrattiniC1G
Fitting, socleC3×C5×C72C2×C3
LCSG ▷ C3×C5×C7
UCSC1 ◁ C7

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 315

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC7×ΓL(1,𝔽8)×D75, C5×C7×ΓL(1,𝔽8)×D15, C7×ΓL(1,𝔽8)×C3⋊(C5⋊D5)
maximal normal subgroup ofC5×C7×ΓL(1,𝔽8)×D15
maximal quotient ofC7×ΓL(1,𝔽8)×D75, C5×C7×ΓL(1,𝔽8)×D15, C7×ΓL(1,𝔽8)×C3⋊(C5⋊D5)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4410, 107);