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×#30392 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#376 ◁ G | |
586 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)(4,5,7,8)(6,9,10,11)(12,13)(14,17)(15,16)(18,19)(20,21)(22,25)(23,27)(24,26)(28,32)(29,33)(30,35)(31,34);(5,8)(9,11)(20,22,28,29,24,31,23,30)(21,25,32,33,26,34,27,35);(4,6,7,10)(5,9,8,11)(13,16)(17,19)(22,29)(23,28)(25,33)(27,32)(30,31)(34,35);(6,10)(9,11)(12,14,15,18)(13,17,16,19)(21,26)(25,34)(27,32)(33,35)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23380);