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C2×C3×ΓL(1,𝔽8)×C31 #16840909

order3906
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C32×C31 · Schur multiplier (H2) = C3

Aliases

direct productC2×C3×ΓL(1,𝔽8)×C31
unambiguous semidirect product(C2×C7)⋊(C32×C31)
unambiguous semidirect product(C2×C3×C7)⋊(C3×C31)

Splittings

Fitting, socleC2×C3×C7×C31 ⋊ C3
derived, SylowC7 ⋊ C2×C32×C31
HallC2×C7×C31 ⋊ C32
HallC7×C31 ⋊ C2×C32
HallC2×C7 ⋊ C32×C31

Subgroups

subgroupG/H
center, hypercenterC2×C3×C31ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C2×C32×C31
FrattiniC1G
Fitting, socleC2×C3×C7×C31C3
LCSG ▷ C7
UCSC1 ◁ C2×C3×C31

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1302

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC2×C3×ΓL(1,𝔽8)×ΓL(1,𝔽32), C3×D5×ΓL(1,𝔽8)×C31, C2×C3×C5×ΓL(1,𝔽8)×C31, …
maximal normal subgroup ofC2×C3×ΓL(1,𝔽8)×ΓL(1,𝔽32), C2×C3×C5×ΓL(1,𝔽8)×C31, C2×C3×C7×ΓL(1,𝔽8)×C31
maximal quotient ofC3×D5×ΓL(1,𝔽8)×C31, C2×C3×C5×ΓL(1,𝔽8)×C31, C2×C3×C31×C49⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3906, 59);