Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×C4×#120 ⋊ C2 |
| Sylow | C3 ⋊ #52103 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C25 |
| Frattini | C23 | C24×D3 |
| Fitting | C3×C4×#120 | C2 |
| hypercenter | C4×#120 | D3 |
| hypocenter | C3 | #52103 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#120 | |
3356 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfg,i^a=i^2,c^b=ch,d^b=df,i^d=i^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #72270671, D3×#52103, #72270696 |
| maximal normal subgroup of | #72270671, D3×#52103, #72270696 |
| maximal quotient of | #72270909 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 804335);