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#90296⋊C3 #13057341

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

Aliases

unambiguous semidirect product#90296⋊C3

Splittings

FittingC7×#121 ⋊ C2×C3
Hall#90296 ⋊ C3
HallΓL(1,𝔽8) ⋊ #534
SylowC7 ⋊ C3×#534

Subgroups

subgroupG/H
center, FrattiniC22C23×AGL(1,𝔽7)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#121C2×C3
hypercenter#121AGL(1,𝔽7)
hypocenterC7C3×#534
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #121

692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#534, #72230275, #72230291, …
maximal normal subgroup ofAGL(1,𝔽7)×#534, #72230275, #72230291, …
maximal quotient of#72259716, #72276654, #72276666

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6630);