Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#11039 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #124 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | #394 |
| prime radical | C2×C3×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5)(6,7)(8,9,12,17,14,10,13,16)(11,18,21,19,22,15,20,23);(8,10)(9,14)(11,19)(12,16)(13,17)(15,21)(18,20)(22,23);(5,7)(8,11)(9,15)(10,18)(12,20)(13,21)(14,22)(16,23)(17,19)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4027);