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C17×C3⋊(C5⋊AGL(1,𝔽7)) #24924222

order10710
automorphism?
homologyAbelianization (H1) = C2×C3×C17 · Schur multiplier (H2) = C1

Aliases

direct productC17×C3⋊(C5⋊AGL(1,𝔽7))
unambiguous semidirect productC3⋊(C17×C5⋊AGL(1,𝔽7))
unambiguous semidirect productC5⋊(C17×C3⋊AGL(1,𝔽7))
unambiguous semidirect productΓL(1,𝔽8)⋊(C17×D15)
unambiguous semidirect productD105⋊(C3×C17)
unambiguous semidirect product(C17×D105)⋊C3

Splittings

Fitting, socleC3×C5×C7×C17 ⋊ C2×C3
derivedC3×C5×C7 ⋊ C2×C3×C17
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C2
HallC3×C17×ΓL(1,𝔽8) ⋊ D5
HallC5×C7×C17 ⋊ C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×C17
HallC7×C17 ⋊ C3×D15
HallC5×C17 ⋊ C3⋊AGL(1,𝔽7)
HallC3×ΓL(1,𝔽8) ⋊ D5×C17
HallC5×C7 ⋊ C3×D3×C17
SylowC7 ⋊ C3×C17×D15
SylowC5 ⋊ C17×C3⋊AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC17C3⋊(C5⋊AGL(1,𝔽7))
commutator, cosocle, hypocenterC3×C5×C7C2×C3×C17
FrattiniC1G
Fitting, socleC3×C5×C7×C17C2×C3
LCSG ▷ C3×C5×C7
UCSC1 ◁ C17

72 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

714 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD17×C3⋊(C5⋊AGL(1,𝔽7)), C17×D15×AGL(1,𝔽7), (C17⋊AGL(1,𝔽7))⋊D15, …
maximal normal subgroup ofD17×C3⋊(C5⋊AGL(1,𝔽7)), C17×D15×AGL(1,𝔽7), (C17⋊AGL(1,𝔽7))⋊D15, …
maximal quotient ofC17×C3⋊(ΓL(1,𝔽8)⋊Dic5), C2×C17×C3⋊(C5⋊AGL(1,𝔽7))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10710, 37);