Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| Fitting | C23×C3×#41 ⋊ C2 |
| Sylow | C3 ⋊ #34736 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C25 |
| Frattini | C23 | C24×D3 |
| Fitting | C23×C3×#41 | C2 |
| hypercenter | C23×#41 | D3 |
| hypocenter | C3 | #34736 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ C23×#41 | |
5758 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|e^2=g,b^a=bf,c^a=cg,c^b=ch,d^b=dh,i^b=i^2156 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#34736, #72253846, #72253858 |
| maximal normal subgroup of | D3×#34736, #72253846, #72253858 |
| maximal quotient of | #72254818, #72270038, #72270860, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 338312);