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2−1+4⋊ΓL(1,𝔽8) #10621616

order672
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product2−1+4⋊ΓL(1,𝔽8)

Splittings

Fitting, HallC7×2−1+4 ⋊ C3
Sylow2−1+4 ⋊ ΓL(1,𝔽8)
SylowC7 ⋊ 2−1+4⋊C3

Subgroups

subgroupG/H
center, FrattiniC2C22×C22⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7×Q8C22×C3
FittingC7×2−1+4C3
socleC2×C7C22×A4
hypercenterD4C22⋊ΓL(1,𝔽8)
LCSG ▷ C7×Q8
UCSC1 ◁ C2 ◁ D4

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Polycyclic presentation:

2^2,3,2^2,7,2|d^2=g,e^2=g,b^a=bg,d^c=e,e^c=deg,f^c=f^2,e^d=eg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13057684, #13057685, #577⋊ΓL(1,𝔽8), …
maximal normal subgroup of#13057684, #13057685, #577⋊ΓL(1,𝔽8), …
maximal quotient of#519⋊ΓL(1,𝔽8), #520⋊ΓL(1,𝔽8), #548⋊ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 1198);