Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #13060786 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, Hall | C7×H(3,𝔽4) ⋊ C3 |
| derived | C24×C7 ⋊ C22×C3 |
| Sylow | H(3,𝔽4) ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ #5450 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C24×C7 | C22×C3 |
| Frattini | C22 | C22×C22⋊ΓL(1,𝔽8) |
| Fitting | C7×H(3,𝔽4) | C3 |
| socle | C22×C7 | C22×A4 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 | |
140 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^2,3,2^4,7|d^a=df,e^a=efg,d^b=dfg,e^b=eg,d^c=de,e^c=d,f^c=g,g^c=fg,h^c=h^455 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#5450, #72277370 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#5450 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10075);