Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#30375 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,8)(9,11)(12,13)(14,17)(15,16)(18,19)(20,21)(22,26)(23,31)(24,27)(25,28)(29,33)(30,34)(32,35);(4,5,7,8)(6,9,10,11)(20,22,29,32,24,23,25,30)(21,26,33,35,27,31,28,34);(4,6)(5,9)(7,10)(8,11)(13,16)(17,19)(20,23,24,22)(21,26,27,31)(25,30,29,32)(28,35,33,34);(6,10)(9,11)(12,14,15,18)(13,17,16,19)(21,27)(26,31)(28,33)(34,35)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23363);