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×#61074 |
| 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 | |
2774 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)(8,9,14,18)(10,20,21,22)(11,19,13,16)(12,17,15,23);(8,10)(9,15)(11,17)(12,18)(13,23)(14,21)(16,20)(19,22);(5,7)(8,11,14,13)(9,16,18,19)(10,17,21,23)(12,22,15,20);(8,12)(9,17)(10,19)(11,22)(13,20)(14,15)(16,21)(18,23);(9,18)(10,21)(16,19)(17,23)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54062);