Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C7 ⋊ #5912 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7×S4 |
| commutator, cosocle | C2×C7×A4 | C23 |
| Fitting | C24×C7 | C2×D3 |
| socle | C23×C7 | C22×D3 |
| hypercenter | C22 | D7×S4 |
| hypocenter | C7×A4 | #41 |
| LCS | G ▷ C2×C7×A4 ▷ C7×A4 | |
| UCS | C1 ◁ C2 ◁ C22 | |
342 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 41 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|a^2=d,f^a=f^6,c^b=cd,e^b=e^2,g^b=gh,g^e=h,h^e=gh70 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11360);