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×#30374 |
| 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 | |
634 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)(23,27)(24,29)(25,28)(30,33)(31,34)(32,35);(4,5,7,8)(6,9,10,11)(20,22,30,32,25,23,24,31)(21,26,33,35,28,27,29,34);(4,6)(5,9)(7,10)(8,11)(13,16)(17,19)(20,23,25,22)(21,27,28,26)(24,31,30,32)(29,34,33,35);(6,10)(9,11)(12,14,15,18)(13,17,16,19)(21,28)(26,27)(29,33)(34,35)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23362);