Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#22055 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
1024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(4,5,7,6)(8,9)(10,14)(11,13)(12,15)(16,17)(18,28)(19,23)(20,27)(21,26)(22,25)(24,30)(29,31);(4,6,7,5)(8,10)(9,12)(11,15)(13,14)(16,18)(17,23)(19,21)(20,30)(22,29)(24,25)(26,28)(27,31);(10,15)(12,14)(16,19,21,18)(17,24,26,31)(20,30,22,29)(23,27,28,25);(9,13)(12,14)(16,20,21,22)(17,25,26,27)(18,29,19,30)(23,31,28,24)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 15043);