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ΓL(1,𝔽8)⋊((C3⋊D3)⋊C3) #11765736

order1134
isoclinismisoclinic class of ΓL(1,𝔽8)⋊((C3⋊D3)⋊C3)
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C32

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊((C3⋊D3)⋊C3)
unambiguous semidirect product#67369⋊C3
unambiguous semidirect product#67385⋊C3
unambiguous semidirect product(C3⋊(C3⋊AGL(1,𝔽7)))⋊C3

Splittings

FittingC7×H(3,𝔽3) ⋊ C2×C3
derivedC32×C7 ⋊ C2×C32
HallΓL(1,𝔽8)×H(3,𝔽3) ⋊ C2
SylowC7 ⋊ C3×(C3⋊D3)⋊C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC32×C7C2×C32
FrattiniC3C3×C3⋊AGL(1,𝔽7)
FittingC7×H(3,𝔽3)C2×C3
socleC3×C7C32×D3
LCSG ▷ C32×C7
UCSC1

116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 63

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×(C3⋊D3)⋊C3, #72224671, #72224677, …
maximal normal subgroup ofAGL(1,𝔽7)×(C3⋊D3)⋊C3, #72224677, #72224685, …
maximal quotient of#72224671, #72230082, C2×ΓL(1,𝔽8)⋊((C3⋊D3)⋊C3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1134, 166);