Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C24×C3×C7 ⋊ 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 | C24×C3×C7 | C22 |
| hypercenter | C24 | D3×D7 |
| hypocenter | C3×C7 | C2×#120 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C24 | |
1324 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 1324 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2^6,3,7|c^a=cf,d^a=de,h^a=h^6,c^b=cf,d^b=de,g^b=g^2132 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7×#120, #72244206, #72244296, … |
| maximal normal subgroup of | D3×D7×#120, #72244206, #72244296 |
| maximal quotient of | #72244285, #72244303, #72246782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 9958);