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ΓL(1,𝔽8)⋊AGL(1,𝔽7) #11725090

order882
isoclinismisoclinic class of ΓL(1,𝔽8)⋊AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊AGL(1,𝔽7)
unambiguous semidirect product(ΓL(1,𝔽8)⋊D7)⋊C3
unambiguous semidirect product#64424⋊C3
unambiguous semidirect product#64425⋊C3

Splittings

Fitting, socle, derived, SylowC72 ⋊ C2×C32
HallΓL(1,𝔽8)2 ⋊ C2
HallC7⋊D7 ⋊ C32

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC72C2×C32
LCSG ▷ C72
UCSC1

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 21

Polycyclic presentation:

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

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10596885⋊C3, (#64418⋊C2)⋊C3, AGL(1,𝔽7)2, …
maximal normal subgroup of#10596885⋊C3, (#64418⋊C2)⋊C3, AGL(1,𝔽7)2, …
maximal quotient ofΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4), C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C5×ΓL(1,𝔽8)⋊AGL(1,𝔽7), …
holomorph ofΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(882, 36);