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#67355⋊C2 #10625426

order756
isoclinismisoclinic class of #67355⋊C2
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#67355⋊C2
unambiguous semidirect productΓL(1,𝔽64)⋊C2
unambiguous semidirect product#67365⋊C2

Splittings

Fitting, derivedC7×C9 ⋊ C22×C3
Hall#4428 ⋊ C22
SylowC7 ⋊ C2×AGL(1,ℤ/9)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC7×C9C22×C3
FrattiniC3D3×AGL(1,𝔽7)
socleC3×C7C2×C3×D3
LCSG ▷ C7×C9
UCSC1

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 63

Polycyclic presentation:

2^2,3^2,7,3|d^3=f,e^a=e^6,d^b=d^2f^2,f^b=f^2,d^c=df,e^c=e^4

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#67355⋊C2, AGL(1,𝔽7)×AGL(1,ℤ/9), #72229933, …
maximal normal subgroup ofC2×#67355⋊C2, AGL(1,𝔽7)×AGL(1,ℤ/9), C3×#67355⋊C2
maximal quotient of#13094201, ΓL(1,𝔽64)⋊C4, #67365⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(756, 57);