Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#23947 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
1270 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)(8,9)(10,14)(11,13)(12,15)(16,17)(18,27)(19,26)(20,25)(21,24)(22,29)(23,28)(30,31);(4,5)(6,7)(8,10,11,15)(9,12,13,14)(16,18)(17,22)(19,30)(20,29)(21,28)(23,26)(24,31)(25,27);(5,7)(16,18)(17,23)(19,28)(20,29)(21,30)(22,26)(24,27)(25,31);(9,13)(12,14)(16,19,20,21)(17,24,25,26)(18,28,29,30)(22,23,27,31)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16935);