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×C4 |
| direct product | C3×#37118 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2539 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2406 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)(5,7)(9,12)(10,14)(19,26)(23,28)(25,30)(29,31);(8,9,11,12)(10,13,14,15)(17,21)(18,24)(23,28)(25,30);(4,5)(6,7)(8,9)(10,13)(11,12)(14,15)(16,17)(18,22)(19,23)(20,21)(24,27)(25,29)(26,28)(30,31);(8,10)(9,13)(11,14)(12,15)(16,18,20,24)(17,22,21,27)(19,25,26,30)(23,29,28,31);(16,19)(17,23)(18,25)(20,26)(21,28)(22,29)(24,30)(27,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30106);