Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#1839 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | C23 |
| commutator | C22 | C23×C3×C8 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle, prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ G | |
1134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(27,28,29);(3,4)(5,10)(6,8)(7,9)(11,12)(13,15)(14,16)(17,18)(19,20,21,23,22,24,25,26);(3,5,7,6)(4,8,9,10)(11,13,14,17)(12,15,16,18);(3,6,7,5)(4,8,9,10)(12,16)(15,18)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13758);