Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#37740 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#538 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(9,12)(13,15)(16,17)(18,21)(19,24)(20,23)(22,27)(25,28)(26,30)(29,31);(5,7)(17,18)(22,24)(23,25)(29,30);(4,5)(6,7)(8,9)(10,13)(11,12)(14,15)(16,18,21,17)(19,24,27,22)(20,25,28,23)(26,30,31,29);(8,10,11,14)(9,13,12,15)(16,19)(17,22)(18,24)(20,26)(21,27)(23,29)(25,30)(28,31);(4,5)(6,7)(16,20,21,28)(17,23,18,25)(19,26,27,31)(22,29,24,30)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30728);