Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#30482 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
1010 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)(8,9,12,18,14,17,13,27)(10,19,22,31,23,30,21,15)(11,24,25,16,26,33,28,32)(20,35,37,39,38,29,36,34);(4,5)(6,7)(8,10,14,23)(9,15,17,31)(11,20,26,38)(12,21,13,22)(16,29,32,35)(18,30,27,19)(24,34,33,39)(25,36,28,37);(5,7)(8,11,14,26)(9,16,17,32)(10,20,23,38)(12,25,13,28)(15,29,31,35)(18,33,27,24)(19,34,30,39)(21,36,22,37);(9,17)(10,21)(12,13)(15,19)(16,32)(20,36)(22,23)(25,28)(29,34)(30,31)(35,39)(37,38)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23470);