Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#37138 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5,8,11)(6,7,10,9)(12,13,15,17)(14,18,19,16)(20,21,24,25)(22,28,30,26)(23,27,31,32)(29,34,35,33);(6,10)(7,9)(21,25)(26,28)(27,32)(33,34);(4,6,8,10)(5,9,11,7)(12,14,15,19)(13,16,17,18)(20,21)(22,26)(23,27)(24,25)(28,30)(29,33)(31,32)(34,35);(4,7,8,9)(5,10,11,6)(20,22,24,30)(21,26,25,28)(23,29,31,35)(27,33,32,34);(4,5)(6,9)(7,10)(8,11)(20,23,24,31)(21,27,25,32)(22,29,30,35)(26,33,28,34)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30126);