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C2×ΓL(1,𝔽8)×3−1+2 #11765764

order1134
automorphism?
homologyAbelianization (H1) = C2×C33 · Schur multiplier (H2) = C32

Aliases

direct productC2×ΓL(1,𝔽8)×3−1+2

Splittings

FittingC2×C7×3−1+2 ⋊ C3
HallC2×C7 ⋊ C3×3−1+2
SylowC7 ⋊ C2×C3×3−1+2

Subgroups

subgroupG/H
centerC2×C3C32×ΓL(1,𝔽8)
commutator, cosocleC3×C7C2×C33
FrattiniC3C2×C32×ΓL(1,𝔽8)
FittingC2×C7×3−1+2C3
socleC2×C3×C7C33
prime radicalC2×C32×ΓL(1,𝔽8)C3
hypercenterC2×3−1+2ΓL(1,𝔽8)
hypocenterC7C2×C3×3−1+2
LCSG ▷ C3×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C2×3−1+2

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 126

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72229769, C2×3−1+2×AGL(1,𝔽7), C2×#72224635, …
maximal normal subgroup ofC2×3−1+2×AGL(1,𝔽7), C2×ΓL(1,𝔽8)×AGL(1,ℤ/9), C2×ΓL(1,𝔽8)⋊AGL(1,ℤ/9), …
maximal quotient of#72229769, C2×#72224635, 3−1+2×#64423, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1134, 194);