Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Sylow | C3 ⋊ #60141 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C24×D3 |
| commutator, socle | C22×C3 | C24×C4 |
| Fitting, prime radical | C2×C3×#525 | C2 |
| cosocle | C2×C3×C4 | C25 |
| hypercenter | C2×#525 | D3 |
| hypocenter | C3 | #60141 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#525 | |
2862 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|a^2=g,c^2=g,g^2=h,b^a=bf,e^a=eh,i^a=i^2,d^c=dh,i^c=i^2144 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#60141, #72249317, #72249319, … |
| maximal normal subgroup of | D3×#60141, #72249317, #72249319, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1033498);