Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#11026 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #121 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | #390 |
| prime radical | C22×C3×#135 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
1278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9)(10,18,21,14)(11,22,25,15)(12,17)(13,16)(19,29)(20,34,26,30)(23,31)(24,35,27,32)(28,33)(36,37)(38,39);(5,7)(8,10)(9,14)(11,23)(12,26)(13,21)(15,31)(16,18)(17,34)(19,24)(20,28)(22,39)(25,38)(27,37)(29,32)(30,33)(35,36);(8,11)(9,15)(10,19)(12,27)(13,25)(14,29)(16,22)(17,35)(18,36)(20,38)(21,37)(23,26)(24,28)(30,39)(31,34)(32,33)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4014);