Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C2×C42 |
| direct product | C3×#7958 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #95 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | C2×#95 |
| prime radical | C24×C3 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C42 ◁ G | |
584 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)(4,5)(6,14)(7,11)(8,12)(9,13)(10,15)(16,19)(17,18)(20,21)(22,30,33,34)(23,27)(24,28)(25,29)(26,32,35,31);(4,6,8,16,9,17,7,15)(5,10,11,18,13,19,12,14)(20,22)(21,26,29,35)(23,31)(24,32)(25,33)(27,34,28,30);(4,7,9,8)(5,11,13,12)(6,15,17,16)(10,18,19,14)(20,23,25,24)(21,27,29,28)(22,31,33,32)(26,34,35,30)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 946);