Finite Group Databasebeta

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#13052040

order1344
automorphism?
homologyAbelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Hall#89912 ⋊ C3
HallΓL(1,𝔽8) ⋊ #504
SylowC7 ⋊ C3×#504

Subgroups

subgroupG/H
center, FrattiniC8C22×AGL(1,𝔽7)
commutator, socleC2×C7C22×C3×C8
FittingC7×#131C2×C3
prime radical#41×ΓL(1,𝔽8)C4
cosocleC7×C8C23×C3
hypercenter#131AGL(1,𝔽7)
hypocenterC7C3×#504
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C8 ◁ #131

168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 224

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 1329);