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C2×C3×ΓL(1,𝔽8)⋊D29 #20300636

order7308
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×ΓL(1,𝔽8)⋊D29
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C3×D29)
unambiguous semidirect product(C2×C29)⋊(C3×AGL(1,𝔽7))
unambiguous semidirect product(C2×D203)⋊C32
unambiguous semidirect product(C2×C3×D203)⋊C3

Splittings

Fitting, socleC2×C3×C7×C29 ⋊ C2×C3
derived, HallC7×C29 ⋊ C22×C32
HallC3×ΓL(1,𝔽8)×C29 ⋊ C22
HallC2×D203 ⋊ C32
HallC3×ΓL(1,𝔽8) ⋊ C2×D29
SylowC29 ⋊ C2×C3×AGL(1,𝔽7)
SylowC7 ⋊ C2×C32×D29

Subgroups

subgroupG/H
center, hypercenterC2×C3ΓL(1,𝔽8)⋊D29
commutator, hypocenterC7×C29C22×C32
FrattiniC1G
Fitting, socleC2×C3×C7×C29C2×C3
LCSG ▷ C7×C29
UCSC1 ◁ C2×C3

120 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

462 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D5×ΓL(1,𝔽8)⋊D29, C2×C3×C5×ΓL(1,𝔽8)⋊D29, C2×C3×C5⋊(ΓL(1,𝔽8)⋊D29)
maximal normal subgroup ofC2×C3×C5×ΓL(1,𝔽8)⋊D29
maximal quotient ofC3×D5×ΓL(1,𝔽8)⋊D29, C2×C3×C5×ΓL(1,𝔽8)⋊D29, C2×C3×C5⋊(ΓL(1,𝔽8)⋊D29)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7308, 111);