Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#17976 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C42 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C3×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,8)(6,10)(9,11)(12,13,17,20,18,14,16,22)(15,21,24,27,25,19,23,26);(4,5,7,9)(6,11,10,8)(12,14,18,13)(15,21,25,19)(16,20,17,22)(23,26,24,27);(12,15,17,24,18,25,16,23)(13,19,20,26,14,21,22,27);(13,14)(19,21)(20,22)(26,27)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10964);