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C7×C3⋊(C5⋊AGL(1,𝔽7)) #17984385

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

Aliases

direct productC7×C3⋊(C5⋊AGL(1,𝔽7))
unambiguous semidirect productC3⋊(C7×C5⋊AGL(1,𝔽7))
unambiguous semidirect productC5⋊(C7×C3⋊AGL(1,𝔽7))
unambiguous semidirect productΓL(1,𝔽8)⋊(C7×D15)
unambiguous semidirect productD105⋊(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 ⋊ C7×C3⋊AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC7C3⋊(C5⋊AGL(1,𝔽7))
commutator, cosocle, hypocenterC3×C5×C7C2×C3×C7
FrattiniC1G
Fitting, socleC3×C5×C72C2×C3
LCSG ▷ C3×C5×C7
UCSC1 ◁ C7

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 630

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC7×C3⋊(ΓL(1,𝔽8)⋊D25), C5×C7×C3⋊(C5⋊AGL(1,𝔽7)), C7×C3⋊(C5⋊(C5⋊AGL(1,𝔽7)))
maximal normal subgroup ofC5×C7×C3⋊(C5⋊AGL(1,𝔽7))
maximal quotient ofC7×C3⋊(ΓL(1,𝔽8)⋊D25), C5×C7×C3⋊(C5⋊AGL(1,𝔽7)), C7×C3⋊(C5⋊(C5⋊AGL(1,𝔽7)))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4410, 106);