Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C3×#21017 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #386 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | C2×#386 |
| prime radical | C3×#3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
1692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,9,15,10,14,8,13)(6,11,17,25,18,26,22,28)(7,19,20,29,21,12,23,27)(16,24,32,35,31,33,30,34);(4,6,9,22,10,18,8,17)(5,11,13,25,14,26,15,28)(7,16,23,32,21,31,20,30)(12,24,27,34,19,33,29,35);(4,7)(5,12)(6,16)(8,20)(9,23)(10,21)(11,24)(13,27)(14,19)(15,29)(17,30)(18,31)(22,32)(25,34)(26,33)(28,35);(4,8,10,9)(5,13,14,15)(6,17,18,22)(7,20,21,23)(11,25,26,28)(12,27,19,29)(16,30,31,32)(24,34,33,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14005);