Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C3×#21006 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1654 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)(8,9,13,19,14,18,12,17)(10,15,23,30,22,31,21,29)(11,24,25,16,26,32,27,33)(20,34,35,28,37,38,36,39);(8,10,12,23,14,22,13,21)(9,15,19,29,18,31,17,30)(11,20,25,36,26,37,27,35)(16,28,24,38,33,39,32,34);(5,7)(8,11)(9,16)(10,20)(12,25)(13,27)(14,26)(15,28)(17,32)(18,33)(19,24)(21,35)(22,37)(23,36)(29,38)(30,34)(31,39);(8,12,14,13)(9,17,18,19)(10,21,22,23)(11,25,26,27)(15,29,31,30)(16,32,33,24)(20,35,37,36)(28,38,39,34)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13994);