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C2×C4×C7×ΓL(1,𝔽8) #11924256

order1176
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C3×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC2×C4×C7×ΓL(1,𝔽8)
unambiguous semidirect product(C2×C4×C7)⋊(C3×C7)

Splittings

Fitting, HallC2×C4×C72 ⋊ C3
derivedC7 ⋊ C2×C3×C4×C7
SylowC72 ⋊ C2×C3×C4

Subgroups

subgroupG/H
center, hypercenterC2×C4×C7ΓL(1,𝔽8)
commutator, hypocenterC7C2×C3×C4×C7
FrattiniC2C22×C7×ΓL(1,𝔽8)
FittingC2×C4×C72C3
socleC22×C72C2×C3
prime radicalC22×C7×ΓL(1,𝔽8)C2
cosocleC2×C7C22×C3×C7
LCSG ▷ C7
UCSC1 ◁ C2×C4×C7

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4×ΓL(1,𝔽8)⋊D7, C2×C7×D4×ΓL(1,𝔽8), C2×C4⋊(ΓL(1,𝔽8)⋊D7), …
maximal normal subgroup ofC2×C4×ΓL(1,𝔽8)⋊D7, C2×C7×D4×ΓL(1,𝔽8), C2×C4⋊(ΓL(1,𝔽8)⋊D7), …
maximal quotient ofC7×M16×ΓL(1,𝔽8), C42×C7×ΓL(1,𝔽8), C7×C22⋊C4×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1176, 171);