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C3×ΓL(1,𝔽8)×D55 #20294176

order6930
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×ΓL(1,𝔽8)×D55
unambiguous semidirect productC5⋊(C3×ΓL(1,𝔽8)×D11)
unambiguous semidirect productC11⋊(C3×D5×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D55)⋊C32
unambiguous semidirect product(C3×C7×D55)⋊C3

Splittings

Fitting, socleC3×C5×C7×C11 ⋊ C2×C3
derived, HallC5×C7×C11 ⋊ C2×C32
HallC3×C5×C11×ΓL(1,𝔽8) ⋊ C2
HallC7×D55 ⋊ C32
HallC3×C11×ΓL(1,𝔽8) ⋊ D5
HallC3×C5×ΓL(1,𝔽8) ⋊ D11
HallC7×C11 ⋊ C32×D5
HallC5×C11 ⋊ C2×C3×ΓL(1,𝔽8)
HallC5×C7 ⋊ C32×D11
SylowC11 ⋊ C3×D5×ΓL(1,𝔽8)
SylowC7 ⋊ C32×D55
SylowC5 ⋊ C3×ΓL(1,𝔽8)×D11

Subgroups

subgroupG/H
center, hypercenterC3ΓL(1,𝔽8)×D55
commutator, hypocenterC5×C7×C11C2×C32
FrattiniC1G
Fitting, socleC3×C5×C7×C11C2×C3
LCSG ▷ C5×C7×C11
UCSC1 ◁ C3

96 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

435 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽7)×D55, C3×D55⋊AGL(1,𝔽7), C3×D5×ΓL(1,𝔽8)×D11, …
maximal normal subgroup ofC3×AGL(1,𝔽7)×D55, C3×D55⋊AGL(1,𝔽7), C3×D5×ΓL(1,𝔽8)×D11, …
maximal quotient ofC3×ΓL(1,𝔽8)×Dic55, C2×C3×ΓL(1,𝔽8)×D55, C3×ΓL(1,𝔽8)×D275, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6930, 63);