Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C3×C7×C4⋊C4 ⋊ C2 |
| Hall | C3×C7 ⋊ #537 |
| Sylow | C7 ⋊ #5510 |
| Sylow | C3 ⋊ #90201 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D21 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C2×C3×C7×C4⋊C4 | C2 |
| hypercenter | C2×C4⋊C4 | D21 |
| hypocenter | C3×C7 | #537 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 | |
588 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 672 |
Polycyclic presentation:
2^6,3,7|b^2=f,d^2=e,c^a=cf,d^a=de,g^a=g^2,h^a=h^6,d^b=df222 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10889);