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ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4) #13183580

order1764
isoclinismisoclinic class of ΓL(1,𝔽8)⋊AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)
unambiguous semidirect product(ΓL(1,𝔽8)⋊Dic7)⋊C3
unambiguous semidirect product#10596874⋊C3
unambiguous semidirect product#10596875⋊C3

Splittings

derived, SylowC72 ⋊ C32×C4
HallΓL(1,𝔽8)2 ⋊ C4
HallC7⋊Dic7 ⋊ C32

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)⋊AGL(1,𝔽7)
commutator, hypocenterC72C32×C4
Fitting, socle, cosocleC2×C72C2×C32
prime radicalC2×ΓL(1,𝔽8)2C2
LCSG ▷ C72
UCSC1 ◁ C2

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 84

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC11×ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4), C11⋊(ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)), C13×ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4), …
maximal normal subgroup ofC11×ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4), C13×ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4), C17×ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4), …
maximal quotient ofC11×ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4), C11⋊(ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)), C13×ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1764, 100);