Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8491 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#8496 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #379 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C3×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(5,9,11,14)(10,18,19,17)(20,21)(22,30,33,35)(23,26)(24,28)(25,29)(27,32,34,31);(4,5)(6,10)(7,14)(8,11)(9,16)(12,17)(13,19)(15,18)(21,26)(23,24)(27,30)(28,29)(31,32)(34,35);(4,6,8,13)(5,10,11,19)(7,15,16,12)(9,17,14,18)(20,22)(21,27)(23,32)(24,31)(25,33)(26,30)(28,35)(29,34)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1484);