Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #89995⋊C3 |
| Fitting | C22×C7×D4 ⋊ C2×C3 |
| Hall | #89995 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #392 |
| Sylow | C7 ⋊ C3×#392 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×AGL(1,𝔽7) |
| commutator, socle, cosocle | C23×C7 | C23×C3 |
| Fitting | C22×C7×D4 | C2×C3 |
| hypercenter | C22×D4 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#392 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
644 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^3,3,2^3,7|b^a=bg,c^a=cf,h^a=h^6,c^b=ce,h^d=h^286 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#392, #72259713, #72259716, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#392, #72259713, #72259716, … |
| maximal quotient of | #72230204, #72276532, #72276608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1412);