Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#61077 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#525 ◁ G | |
2762 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)(10,19)(11,18)(12,16)(13,15)(14,17)(20,23)(21,22);(8,10)(9,12)(11,16)(13,22)(14,20)(15,19)(17,21)(18,23);(5,7)(8,11,14,13)(9,15,17,18)(10,16,20,22)(12,19,21,23);(8,12,14,21)(9,16,17,22)(10,15,20,18)(11,19,13,23);(9,17)(10,20)(15,18)(16,22)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54065);