Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#17562 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C22×C22⋊C4 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C42 ◁ G | |
992 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)(6,12,14,9)(7,10)(8,11)(13,18,19,16)(15,17)(20,21)(22,28)(23,31)(24,26)(25,29)(27,33)(30,35)(32,34);(4,6,8,14)(5,9,11,12)(7,13,15,19)(10,16,17,18)(20,22,24,30)(21,25,26,34)(23,32,33,29)(27,35,31,28);(21,26)(25,34)(27,31)(28,35);(4,7,8,15)(5,10,11,17)(6,13,14,19)(9,16,12,18)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10550);