Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#24400 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1322 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)(8,9,12,17,14,19,13,18)(10,15,21,30,22,31,28,35)(11,23,25,34,27,33,26,16)(20,32,24,38,36,39,37,29);(5,7)(8,10)(9,15)(11,24)(12,21)(13,28)(14,22)(16,32)(17,30)(18,35)(19,31)(20,26)(23,38)(25,36)(27,37)(29,33)(34,39);(8,11,14,27)(9,16,19,34)(10,20,22,36)(12,26,13,25)(15,29,31,38)(17,33,18,23)(21,37,28,24)(30,39,35,32);(5,7)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 17388);