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×#37113 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1518 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)(4,5,8,9)(6,7,11,10)(12,13,16,15)(14,18,19,17)(23,30)(27,32)(29,34)(33,35);(5,9)(7,10)(21,25)(22,28)(27,32)(29,34);(4,6)(5,10)(7,9)(8,11)(12,14,16,19)(13,17,15,18)(20,21,24,25)(22,26,28,31)(23,27,30,32)(29,33,34,35);(4,7,8,10)(5,6,9,11)(20,22)(21,26)(23,29)(24,28)(25,31)(27,33)(30,34)(32,35);(4,6)(5,10)(7,9)(8,11)(12,15,16,13)(14,18,19,17)(20,23)(21,27)(22,29)(24,30)(25,32)(26,33)(28,34)(31,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30101);