Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| Fitting | C3×#1440 ⋊ C2 |
| Sylow | C3 ⋊ #13839 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator | C23×C3 | C23×C4 |
| Fitting | C3×#1440 | C2 |
| socle, cosocle | C24×C3 | C24 |
| prime radical | C23×#67 | C22 |
| hypercenter | #1440 | D3 |
| hypocenter | C3 | #13839 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ #1440 | |
2702 conjugacy classes of subgroups.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|a^2=h,d^2=e,b^a=be,c^a=cf,c^b=cg,i^c=i^2,i^d=i^2132 conjugacy classes — view the full character table.
| maximal subgroup of | #72270930, #72270989, #72271006, … |
| maximal normal subgroup of | #72270930, #72270989, #72271006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 91245);