Finite Group Databasebeta

← Prev (#17984379) Next (#17984381) →

C5×C3⋊(ΓL(1,𝔽8)⋊D7) #17984380

order4410
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×C3⋊(ΓL(1,𝔽8)⋊D7)
unambiguous semidirect productC3⋊(C5×ΓL(1,𝔽8)⋊D7)
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×D21)

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC5C3⋊(ΓL(1,𝔽8)⋊D7)
commutator, cosocle, hypocenterC3×C72C2×C3×C5
FrattiniC1G
Fitting, socleC3×C5×C72C2×C3
LCSG ▷ C3×C72
UCSC1 ◁ C5

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 315

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC25×C3⋊(ΓL(1,𝔽8)⋊D7), C52×C3⋊(ΓL(1,𝔽8)⋊D7), C5×C3⋊(C5⋊(ΓL(1,𝔽8)⋊D7))
maximal normal subgroup ofC25×C3⋊(ΓL(1,𝔽8)⋊D7), C52×C3⋊(ΓL(1,𝔽8)⋊D7)
maximal quotient ofC25×C3⋊(ΓL(1,𝔽8)⋊D7), C52×C3⋊(ΓL(1,𝔽8)⋊D7), C5×C3⋊(C5⋊(ΓL(1,𝔽8)⋊D7))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4410, 101);