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C2×C3⋊(ΓL(1,𝔽8)⋊D29) #20300639

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

Aliases

direct productC2×C3⋊(ΓL(1,𝔽8)⋊D29)
unambiguous semidirect product(C2×C3)⋊(ΓL(1,𝔽8)⋊D29)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊D87
unambiguous semidirect product(C2×C29)⋊(C3⋊AGL(1,𝔽7))
unambiguous semidirect product(C2×D609)⋊C3

Splittings

Fitting, socleC2×C3×C7×C29 ⋊ C2×C3
derivedC3×C7×C29 ⋊ C22×C3
HallC3×ΓL(1,𝔽8)×C29 ⋊ C22
HallC7×C29 ⋊ C2×C3×D3
HallC3×ΓL(1,𝔽8) ⋊ C2×D29
SylowC29 ⋊ C2×C3⋊AGL(1,𝔽7)
SylowC7 ⋊ C2×C3×D87

Subgroups

subgroupG/H
center, hypercenterC2C3⋊(ΓL(1,𝔽8)⋊D29)
commutator, hypocenterC3×C7×C29C22×C3
FrattiniC1G
Fitting, socleC2×C3×C7×C29C2×C3
LCSG ▷ C3×C7×C29
UCSC1 ◁ C2

88 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

444 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×C3⋊(ΓL(1,𝔽8)⋊D29), C2×C5×C3⋊(ΓL(1,𝔽8)⋊D29), C2×C3⋊(C5⋊(ΓL(1,𝔽8)⋊D29))
maximal normal subgroup ofC2×C5×C3⋊(ΓL(1,𝔽8)⋊D29)
maximal quotient ofD5×C3⋊(ΓL(1,𝔽8)⋊D29), C2×C5×C3⋊(ΓL(1,𝔽8)⋊D29), C2×C3⋊(C5⋊(ΓL(1,𝔽8)⋊D29))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7308, 114);