Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#9730 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | #121 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3)(5,7)(8,9,11,13,12,15,10,14)(16,17)(18,23)(19,28)(20,30)(21,25)(22,26)(24,29)(27,31);(4,5,6,7)(9,13)(10,11)(14,15)(16,18,20,26,21,27,29,19)(17,22,24,23,25,28,30,31);(16,19,29,27,21,26,20,18)(17,23,30,22,25,31,24,28)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2718);