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#127⋊ΓL(1,𝔽8) #10621176

order672
isoclinismisoclinic class of #127⋊ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C4

Aliases

unambiguous semidirect product#127⋊ΓL(1,𝔽8)

Splittings

Fitting, HallC7×#127 ⋊ C3
derivedC42×C7 ⋊ C2×C3
Sylow#127 ⋊ ΓL(1,𝔽8)
SylowC7 ⋊ #127⋊C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC42×C7C2×C3
FrattiniC22C2×C22⋊ΓL(1,𝔽8)
FittingC7×#127C3
socleC22×C7C2×A4
LCSG ▷ C42×C7
UCSC1

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 84

Polycyclic presentation:

2,3,2^2,7,2^2|c^2=f,d^2=g,c^a=cf,d^a=dg,c^b=df,d^b=cd,e^b=e^2,f^b=g,g^b=fg

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7⋊(#127⋊C3), #13057651, C2×#127⋊ΓL(1,𝔽8), …
maximal normal subgroup ofD7⋊(#127⋊C3), #13057651, C2×#127⋊ΓL(1,𝔽8), …
maximal quotient of#384⋊ΓL(1,𝔽8), #392⋊ΓL(1,𝔽8), #399⋊ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 758);