Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#1813 ⋊ C2 |
| Sylow | C3 ⋊ #24288 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #5525 |
| commutator, cosocle | C22×C3×C4 | C24 |
| Frattini | C22×C4 | C23×D3 |
| Fitting | C3×#1813 | C2 |
| socle | C2×C3 | #2408 |
| hypercenter | #1813 | D3 |
| hypocenter | C3 | #24288 |
| LCS | G ▷ C22×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #1813 | |
1290 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|b^2=e,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gh,i^a=i^2,c^b=cg,f^b=fh,d^c=dh,f^c=fh,i^c=i^272 conjugacy classes — view the full character table.
| maximal subgroup of | #72270212, #72270233, #72270246, … |
| maximal normal subgroup of | #72270212, #72270233, #72270246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 205512);