Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#9417 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C3×#3184 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
442 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)(4,5,10,13)(6,9,12,8)(7,14)(11,15)(16,18)(17,19)(20,21)(22,29,30,33)(23,26,31,32)(24,27)(25,28)(34,35);(4,6,8,5,10,12,9,13)(7,14,16,11,17,19,15,18)(20,22,24,32,25,30,34,26)(21,23,27,33,28,31,35,29);(4,7)(5,11)(6,14)(8,16)(9,15)(10,17)(12,19)(13,18)(20,23)(21,26)(22,27)(24,33)(25,31)(28,32)(29,34)(30,35)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2405);