Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10046 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10059 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
570 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)(8,9,12,15)(10,14,11,13)(16,17,22,26)(18,27,19,30)(20,25,21,24)(23,29,31,28);(4,5)(6,7)(9,13)(10,11)(14,15)(16,18,22,19)(17,23,26,31)(20,29,21,28)(24,30,25,27);(5,7)(8,10,12,11)(9,14,15,13)(16,19,22,18)(17,23,26,31)(20,28,21,29)(24,30,25,27)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3047);