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C2×C3×C7×C17×ΓL(1,𝔽8) #28418426

order14994
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C32×C7×C17 · Schur multiplier (H2) = C3

Aliases

direct productC2×C3×C7×C17×ΓL(1,𝔽8)
unambiguous semidirect product(C2×C7)⋊(C32×C7×C17)
unambiguous semidirect product(C2×C3×C7)⋊(C3×C7×C17)
unambiguous semidirect product(C2×C7×C17)⋊(C32×C7)
unambiguous semidirect product(C2×C3×C7×C17)⋊(C3×C7)

Splittings

Fitting, socleC2×C3×C72×C17 ⋊ C3
derivedC7 ⋊ C2×C32×C7×C17
HallC2×C72×C17 ⋊ C32
HallC72×C17 ⋊ C2×C32
HallC2×C72 ⋊ C32×C17
SylowC72 ⋊ C2×C32×C17

Subgroups

subgroupG/H
center, hypercenterC2×C3×C7×C17ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C2×C32×C7×C17
FrattiniC1G
Fitting, socleC2×C3×C72×C17C3
LCSG ▷ C7
UCSC1 ◁ C2×C3×C7×C17

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2142

Conjugacy classes & character table

3570 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3×C7×C17×ΓL(1,𝔽8)⋊C4, C3×C7×ΓL(1,𝔽8)⋊Dic17, C3×C7×ΓL(1,𝔽8)×Dic17, …
maximal normal subgroup ofC3×C7×C17×ΓL(1,𝔽8)⋊C4, C3×C7×ΓL(1,𝔽8)⋊Dic17, C3×C7×ΓL(1,𝔽8)×Dic17, …
maximal quotient ofC3×C4×C7×C17×ΓL(1,𝔽8), C22×C3×C7×C17×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14994, 118);