Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #13830 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator | C22×C3 | C22×C42 |
| Fitting | C2×C3×#384 | C2 |
| socle, prime radical, cosocle | C24×C3 | C24 |
| hypercenter | C2×#384 | D3 |
| hypocenter | C3 | #13830 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C2×#384 | |
1326 conjugacy classes of subgroups; only the 491 normal subgroups are shown.
This group has 1326 conjugacy classes of subgroups — too many to draw. Only its 491 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 768 |
Polycyclic presentation:
2^8,3|a^2=g,b^2=h,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=def,i^a=i^2,c^b=cef,d^b=de132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 91195);