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#90468⋊C3 #13057511

order1344
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

unambiguous semidirect product#90468⋊C3

Splittings

FittingC7×#137 ⋊ C2×C3
Hall#90468 ⋊ C3
HallΓL(1,𝔽8) ⋊ #575
SylowC7 ⋊ C3×#575

Subgroups

subgroupG/H
centerC2C2×D4×AGL(1,𝔽7)
commutator, cosocleC4×C7C24×C3
FrattiniC4C23×AGL(1,𝔽7)
FittingC7×#137C2×C3
socleC2×C7C22×C3×D4
hypercenter#137AGL(1,𝔽7)
hypocenterC7C3×#575
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Polycyclic presentation:

2^4,3,2^2,7|d^2=g,f^2=g,d^a=dg,h^a=h^6,c^b=cg,d^c=df,f^c=fg,f^d=fg,h^e=h^2

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#575, #72229450, #72229639, …
maximal normal subgroup ofAGL(1,𝔽7)×#575, #72229450, #72229639, …
maximal quotient of#72218555, #72218565, #72218610

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6800);