Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C3×#20612 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×#120 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | C2×#379 |
| prime radical | C2×C3×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
3640 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)(4,5,8,13,9,12,21,26)(6,14,17,27,16,10,20,23)(7,11,19,25,18,24,31,34)(15,22,30,33,29,28,32,35);(4,6)(5,10)(7,15)(8,20)(9,16)(11,22)(12,14)(13,27)(17,21)(18,29)(19,32)(23,26)(24,28)(25,35)(30,31)(33,34);(4,7)(5,11)(6,15)(8,19)(9,18)(10,22)(12,24)(13,25)(14,28)(16,29)(17,30)(20,32)(21,31)(23,33)(26,34)(27,35);(5,12)(10,14)(11,24)(13,26)(22,28)(23,27)(25,34)(33,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13600);