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C2×C3×ΓL(1,𝔽8)×D29 #20300637

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

Aliases

direct productC2×C3×ΓL(1,𝔽8)×D29
unambiguous semidirect productC29⋊(C22×C3×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C29)⋊(C2×C3×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C7×D29)⋊C32
unambiguous semidirect product(C2×C3×C7×D29)⋊C3

Splittings

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

480 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×ΓL(1,𝔽8)×D145
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×ΓL(1,𝔽8)×D145

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7308, 112);