Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#29638 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
1292 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)(10,12)(11,14)(13,15)(16,17,20,23)(18,24,26,21)(19,22,27,29)(25,30,31,28);(4,5)(6,7)(16,18,20,26)(17,21,23,24)(19,25,27,31)(22,28,29,30);(5,7)(9,12)(10,15)(13,14)(16,19)(17,22)(18,25)(20,27)(21,28)(23,29)(24,30)(26,31);(19,27)(22,29)(25,31)(28,30)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22626);