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C7×D5×C11×ΓL(1,𝔽8) #30633547

order16170
automorphism?
homologyAbelianization (H1) = C2×C3×C7×C11 · Schur multiplier (H2) = C1

Aliases

direct productC7×D5×C11×ΓL(1,𝔽8)
unambiguous semidirect productC5⋊(C2×C7×C11×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D5)⋊(C3×C7×C11)
unambiguous semidirect product(C7×D5×C11)⋊(C3×C7)

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC7×C11D5×ΓL(1,𝔽8)
commutator, cosocle, hypocenterC5×C7C2×C3×C7×C11
FrattiniC1G
Fitting, socleC5×C72×C11C2×C3
LCSG ▷ C5×C7
UCSC1 ◁ C7×C11

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1155

Conjugacy classes & character table

1540 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC7×C11×AGL(1,𝔽5)×ΓL(1,𝔽8), C7×ΓL(1,𝔽8)×C11⋊AGL(1,𝔽5), C7×C11×ΓL(1,𝔽8)⋊AGL(1,𝔽5), …
maximal normal subgroup ofC7×C11×AGL(1,𝔽5)×ΓL(1,𝔽8), C7×ΓL(1,𝔽8)×C11⋊AGL(1,𝔽5), C7×C11×ΓL(1,𝔽8)⋊AGL(1,𝔽5), …
maximal quotient ofC7×C11×Dic5×ΓL(1,𝔽8), C2×C7×D5×C11×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16170, 92);