Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#30488 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
982 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)(8,9,13,19,15,21,14,20)(10,22,24,35,26,34,25,16)(11,17,28,36,23,33,30,37)(12,29,27,38,31,39,32,18);(8,10,15,26)(9,16,21,35)(11,27,23,32)(12,30,31,28)(13,25,14,24)(17,29,33,39)(18,37,38,36)(19,34,20,22);(5,7)(8,11,15,23)(9,17,21,33)(10,12,26,31)(13,28,14,30)(16,18,35,38)(19,36,20,37)(22,29,34,39)(24,27,25,32);(5,7)(8,12)(9,18)(10,23)(11,26)(13,32)(14,27)(15,31)(16,33)(17,35)(19,39)(20,29)(21,38)(22,36)(24,28)(25,30)(34,37)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23476);