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(C3⋊AGL(1,𝔽7))⋊D7 #13183641

order1764
isoclinismisoclinic class of (C3⋊AGL(1,𝔽7))⋊D7
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product(C3⋊AGL(1,𝔽7))⋊D7
unambiguous semidirect product(ΓL(1,𝔽8)⋊D7)⋊D3

Splittings

Fitting, socle, derivedC3×C72 ⋊ C22×C3
HallC3×C7×ΓL(1,𝔽8) ⋊ C22
SylowC72 ⋊ C2×C3×D3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C72C22×C3
LCSG ▷ C3×C72
UCSC1

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 126

Polycyclic presentation:

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC11×(C3⋊AGL(1,𝔽7))⋊D7, (ΓL(1,𝔽8)⋊D7)⋊D33, (C3⋊AGL(1,𝔽7))⋊D77, …
maximal normal subgroup ofC11×(C3⋊AGL(1,𝔽7))⋊D7, C13×(C3⋊AGL(1,𝔽7))⋊D7, C17×(C3⋊AGL(1,𝔽7))⋊D7, …
maximal quotient ofC11×(C3⋊AGL(1,𝔽7))⋊D7, (ΓL(1,𝔽8)⋊D7)⋊D33, (C3⋊AGL(1,𝔽7))⋊D77, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1764, 161);