Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#21098 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×M16 ◁ G | |
1022 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)(8,9,13,17,14,18,12,28)(10,19,23,30,21,33,22,15)(11,24,26,31,27,38,25,16)(20,34,37,39,35,29,36,32);(8,10)(9,15,18,30)(11,20)(12,22)(13,23)(14,21)(16,29,31,34)(17,19,28,33)(24,32,38,39)(25,36)(26,37)(27,35);(5,7)(8,11,12,25,14,27,13,26)(9,16,17,24,18,31,28,38)(10,20,22,36,21,35,23,37)(15,29,19,32,30,34,33,39);(10,21)(15,30)(19,33)(20,35)(22,23)(29,34)(32,39)(36,37)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14086);