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ΓL(1,𝔽8)×#214 #11759100

order1008
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×#214
unambiguous semidirect product(C7×#214)⋊C3

Splittings

HallC3×ΓL(1,𝔽8) ⋊ SD16
SylowC7 ⋊ C3×#214

Subgroups

subgroupG/H
centerC2ΓL(1,𝔽8)×#67
commutator, cosocleC3×C4×C7C22×C3
FrattiniC4C2×D3×ΓL(1,𝔽8)
FittingC3×C7×D4C2×C3
socleC2×C3×C7C3×D4
prime radicalC3×D4×ΓL(1,𝔽8)C2
hypercenterD4D3×ΓL(1,𝔽8)
hypocenterC3×C7C3×SD16
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C4 ◁ D4

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#932, AGL(1,𝔽7)×#214, #72245544, …
maximal normal subgroup ofΓL(1,𝔽8)×#932, AGL(1,𝔽7)×#214, #72245544, …
maximal quotient ofC3×ΓL(1,𝔽8)×#214

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1008, 271);