Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#11013 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1262 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)(11,15)(12,17)(13,16)(19,36,37,28)(20,34,25,29)(22,39,38,30)(23,32)(24,31)(26,35)(27,33);(8,10)(9,14)(11,22)(12,25)(13,21)(15,30)(16,18)(17,34)(19,23)(20,27)(24,38)(26,37)(28,32)(29,33)(31,39)(35,36);(5,7)(8,11,13,24)(9,15,16,31)(10,19,21,37)(12,26,27,23)(14,28,18,36)(17,35,33,32)(20,38,25,22)(29,39,34,30)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4001);