Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C3×C7×C22⋊C4 ⋊ C22 |
| Hall | C3×C7 ⋊ C2×#120 |
| Sylow | C7 ⋊ D3×#120 |
| Sylow | C3 ⋊ D7×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3×D7 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C7×C22⋊C4 | C22 |
| hypercenter | C22⋊C4 | D3×D7 |
| hypocenter | C3×C7 | C2×#120 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
1324 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2^6,3,7|d^2=f,d^a=df,h^a=h^6,d^b=df,g^b=g^2,d^c=de114 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7×#120, #72244221, #72244304, … |
| maximal normal subgroup of | D3×D7×#120, #72244221, #72244304 |
| maximal quotient of | #72244304, #72244315, #72244353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 7766);