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C17×ΓL(1,𝔽8)×D15 #24924224

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

Aliases

direct productC17×ΓL(1,𝔽8)×D15
unambiguous semidirect productC3⋊(D5×C17×ΓL(1,𝔽8))
unambiguous semidirect productC5⋊(D3×C17×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D15)⋊(C3×C17)
unambiguous semidirect product(C7×C17×D15)⋊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 ⋊ D3×ΓL(1,𝔽8)
HallC3×ΓL(1,𝔽8) ⋊ D5×C17
HallC5×C7 ⋊ C3×D3×C17
SylowC7 ⋊ C3×C17×D15
SylowC5 ⋊ D3×C17×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC17ΓL(1,𝔽8)×D15
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

765 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC17×D15×AGL(1,𝔽7), D15×C17⋊AGL(1,𝔽7), ΓL(1,𝔽8)×D15×D17, …
maximal normal subgroup ofC17×D15×AGL(1,𝔽7), D15×C17⋊AGL(1,𝔽7), ΓL(1,𝔽8)×D15×D17, …
maximal quotient ofC17×ΓL(1,𝔽8)×Dic15, C2×C17×ΓL(1,𝔽8)×D15

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10710, 39);