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C7×ΓL(1,𝔽8)×D55 #30633548

order16170
automorphism?
homologyAbelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×ΓL(1,𝔽8)×D55
unambiguous semidirect productC5⋊(C7×ΓL(1,𝔽8)×D11)
unambiguous semidirect productC11⋊(C7×D5×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D55)⋊(C3×C7)

Splittings

Fitting, socle, HallC5×C72×C11 ⋊ C2×C3
derivedC5×C7×C11 ⋊ C2×C3×C7
HallC5×C7×C11×ΓL(1,𝔽8) ⋊ C2
HallC72×D55 ⋊ C3
HallC7×C11×ΓL(1,𝔽8) ⋊ D5
HallC5×C7×ΓL(1,𝔽8) ⋊ D11
HallC72×C11 ⋊ C3×D5
HallC5×C72 ⋊ C3×D11
HallC5×C11 ⋊ C2×C7×ΓL(1,𝔽8)
SylowC72 ⋊ C3×D55
SylowC11 ⋊ C7×D5×ΓL(1,𝔽8)
SylowC5 ⋊ C7×ΓL(1,𝔽8)×D11

Subgroups

subgroupG/H
center, hypercenterC7ΓL(1,𝔽8)×D55
commutator, cosocle, hypocenterC5×C7×C11C2×C3×C7
FrattiniC1G
Fitting, socleC5×C72×C11C2×C3
LCSG ▷ C5×C7×C11
UCSC1 ◁ C7

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1155

Conjugacy classes & character table

1015 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC7×D5×ΓL(1,𝔽8)×D11, C7×AGL(1,𝔽7)×D55, C7×D55⋊AGL(1,𝔽7), …
maximal normal subgroup ofC7×D5×ΓL(1,𝔽8)×D11, C7×AGL(1,𝔽7)×D55, C7×D55⋊AGL(1,𝔽7), …
maximal quotient ofC7×ΓL(1,𝔽8)×Dic55, C2×C7×ΓL(1,𝔽8)×D55

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16170, 93);