Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#23649 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×#124 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | C2×#397 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,14,17,27)(7,11)(8,12)(9,13)(10,16,24,21)(15,30,18,29)(19,25)(20,26)(22,28)(23,32,34,31)(33,35);(4,6)(5,10)(7,18)(8,21)(9,17)(11,23)(12,27)(13,24)(14,28)(15,19)(16,22)(20,31)(25,34)(26,30)(29,35)(32,33);(4,7,8,20,9,19,22,33)(5,11,12,26,13,25,28,35)(6,15,16,31,17,18,21,32)(10,23,14,29,24,34,27,30);(7,19)(11,25)(15,18)(20,33)(23,34)(26,35)(29,30)(31,32)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16637);