Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#39876 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(4,5,6,7)(8,9)(10,11)(12,13)(14,15);(9,13)(11,14)(17,21)(19,22);(4,5)(6,7)(16,17)(18,22)(19,23)(20,21);(8,10)(9,14)(11,13)(12,15)(16,18,20,23)(17,22,21,19);(8,11,12,14)(9,10,13,15)(16,19,20,22)(17,23,21,18)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 32864);