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#90277⋊C3 #13057320

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

Aliases

unambiguous semidirect product#90277⋊C3

Splittings

FittingC7×#120 ⋊ C2×C3
Hall#90277 ⋊ C3
HallΓL(1,𝔽8) ⋊ #525
SylowC7 ⋊ C3×#525

Subgroups

subgroupG/H
center, FrattiniC22C23×AGL(1,𝔽7)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#120C2×C3
hypercenter#120AGL(1,𝔽7)
hypocenterC7C3×#525
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #120

660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#525, #72230307, #72277502, …
maximal normal subgroup ofAGL(1,𝔽7)×#525, #72230307, #72277502, …
maximal quotient of#120×ΓL(1,𝔽8)⋊C4, #72218604

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6609);