Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C7×2−1+4 ⋊ C2×C3 |
| Hall | #90529 ⋊ C3 |
| Sylow | C7 ⋊ #578⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | A4×C22⋊D7 |
| commutator, cosocle | C7×#41 | C22×C3 |
| Frattini | C4 | C2×A4×D7 |
| Fitting | C7×2−1+4 | C2×C3 |
| socle | C2×C7 | D4×A4 |
| hypercenter | D4 | A4×D7 |
| hypocenter | C7×Q8 | C3×D4 |
| LCS | G ▷ C7×#41 ▷ C7×Q8 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
146 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 29 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:
2^2,3,2^3,7,2|d^2=h,e^2=h,f^2=h,b^a=bd,d^a=dh,g^a=g^6,d^b=dh,e^c=ef,f^c=e,f^e=fh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6915);