Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#33521 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C22×C3×#135 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,14,19)(10,20,23,37)(11,18,13,16)(12,25,28,33)(15,30,32,22)(17,24,35,27)(21,38,29,34)(26,36,39,31);(8,10)(9,15)(11,21)(12,26)(13,29)(14,23)(16,31)(17,34)(18,36)(19,32)(20,25)(22,27)(24,30)(28,39)(33,37)(35,38);(5,7);(8,11,14,13)(9,16,19,18)(10,21,23,29)(12,24,28,27)(15,31,32,36)(17,33,35,25)(20,34,37,38)(22,26,30,39)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26509);