Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| Fitting | C22×C3×#121 ⋊ C2 |
| Sylow | C3 ⋊ #15124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator, socle, cosocle | C24×C3 | C24 |
| Fitting | C22×C3×#121 | C2 |
| hypercenter | C22×#121 | D3 |
| hypocenter | C3 | #15124 |
| LCS | G ▷ C24×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×#121 | |
3822 conjugacy classes of subgroups.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|b^2=e,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,i^b=i^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #72270809, #72270818, #72270846, … |
| maximal normal subgroup of | #72270809, #72270818, #72270846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 106112);