Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×C3×#2736 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C26×C3 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
3382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(27,28,29);(3,4)(5,9)(6,7)(8,10)(11,12,15,16)(13,17,18,14)(19,20,22,24)(21,25,26,23);(4,7)(5,10)(12,16)(14,17)(19,20,22,24)(21,23,26,25);(3,5)(4,8)(6,10)(7,9)(11,13)(12,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,26);(3,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,17)(16,18)(19,21)(20,23)(22,26)(24,25)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 45159);