Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×C4⋊#95 ⋊ C2 |
| Sylow | C3 ⋊ D4×#95 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator | C22×C3 | C22×C42 |
| Fitting | C3×C4⋊#95 | C2 |
| socle, cosocle | C24×C3 | C24 |
| prime radical | C23×D12 | C22 |
| hypercenter | C4⋊#95 | D3 |
| hypocenter | C3 | D4×#95 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C4⋊#95 | |
3562 conjugacy classes of subgroups.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|a^2=g,b^2=h,d^2=g,b^a=be,c^a=cf,d^b=de,i^c=i^2,i^d=i^2156 conjugacy classes — view the full character table.
| maximal subgroup of | #72270789, #72270812, #72270818, … |
| maximal normal subgroup of | #72270789, #72270812, #72270818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 91037);