Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C22×C7×D4 ⋊ D3 |
| Sylow | C7 ⋊ #5415 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10621506 |
| commutator, cosocle | C4×C7×A4 | C22 |
| Frattini | C4 | C2×C7⋊S4 |
| Fitting | C22×C7×D4 | D3 |
| socle | C23×C7 | #67 |
| hypercenter | D4 | C7⋊S4 |
| hypocenter | C7×A4 | D8 |
| LCS | G ▷ C4×C7×A4 ▷ C2×C7×A4 ▷ C7×A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
174 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 27 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:
2^4,3,7,2^2|c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^6,g^a=gh,c^b=cd,g^e=h,h^e=gh79 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6472);