Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C3×#21012 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #386 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | C2×#386 |
| prime radical | C3×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
1732 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)(4,5,9,14,10,15,8,13)(6,16,20,27,19,26,18,11)(7,21,22,29,24,12,23,28)(17,30,31,25,33,35,32,34);(4,6)(5,11,15,27)(7,17)(8,18)(9,20)(10,19)(12,25,21,34)(13,26,14,16)(22,31)(23,32)(24,33)(28,35,29,30);(4,7)(5,12)(6,17)(8,22)(9,23)(10,24)(11,25)(13,28)(14,29)(15,21)(16,30)(18,31)(19,33)(20,32)(26,35)(27,34);(4,8,10,9)(5,13,15,14)(6,18,19,20)(7,22,24,23)(11,26,27,16)(12,28,21,29)(17,31,33,32)(25,35,34,30)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14000);