Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C3×#20952 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #120 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×#120 |
| prime radical | C2×C3×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
1818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(5,7)(8,9,13,17,14,16,12,15)(10,18,19,11,21,22,20,23);(4,5,6,7)(8,10)(9,11)(12,19)(13,20)(14,21)(15,22)(16,23)(17,18);(8,11)(9,10)(12,22)(13,18)(14,23)(15,19)(16,21)(17,20);(8,12,14,13)(9,15,16,17)(10,19,21,20)(11,22,23,18)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13940);