Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#17385 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C22⋊C4 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C22×C22⋊C4 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C42 ◁ G | |
988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,14,17,10)(7,11)(8,12)(9,13)(15,19,16,18)(20,21,24,28,25,29,23,27)(22,26,32,35,30,33,31,34);(4,6,9,17)(5,10,13,14)(7,15,8,16)(11,18,12,19)(20,22)(21,26)(23,31)(24,32)(25,30)(27,34)(28,35)(29,33);(4,7,9,8)(5,11,13,12)(6,15,17,16)(10,18,14,19)(22,30)(26,33)(31,32)(34,35);(4,8,9,7)(5,12,13,11)(6,16,17,15)(10,19,14,18)(20,23,25,24)(21,27,29,28)(22,31,30,32)(26,34,33,35)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10373);