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×#23916 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3487 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1694 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)(10,18,20,32)(11,15,22,30)(12,24,25,16)(14,26,29,21)(19,35,36,34)(23,33,27,31)(28,38,39,37);(4,5)(6,7)(11,22)(12,25)(15,30)(16,24)(18,32)(19,36)(21,26)(28,39);(5,7)(8,10)(9,14)(11,19)(12,26)(13,20)(15,28)(16,32)(17,29)(18,24)(21,25)(22,36)(23,38)(27,37)(30,39)(31,34)(33,35);(8,11,13,22)(9,15,17,30)(10,19,20,36)(12,27,25,23)(14,28,29,39)(16,33,24,31)(18,34,32,35)(21,38,26,37)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16904);