Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#23914 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1706 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)(8,9,13,17)(10,14,19,29)(11,21,22,33)(12,24,27,16)(15,26,31,23)(18,34,35,38)(20,32,25,30)(28,37,39,36);(4,5)(6,7)(8,10,13,19)(9,14,17,29)(11,18,22,35)(12,25,27,20)(15,28,31,39)(16,32,24,30)(21,38,33,34)(23,37,26,36);(5,7)(8,11)(9,15)(10,18)(12,26)(13,22)(14,28)(16,33)(17,31)(19,35)(20,37)(21,24)(23,27)(25,36)(29,39)(30,38)(32,34);(10,19)(12,27)(14,29)(16,24)(18,35)(21,33)(23,26)(28,39)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16902);