Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#26835 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G | |
846 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)(10,15)(11,13)(12,14)(16,17)(18,23)(19,29)(20,30)(21,25)(22,27)(24,26)(28,31);(4,5)(6,7)(8,10,12,11)(9,13,14,15)(16,18,26,19,21,28,20,27)(17,22,30,31,25,29,24,23);(5,7)(8,11,12,10)(9,13,14,15)(16,18,20,27,21,28,26,19)(17,22,24,23,25,29,30,31);(9,14)(13,15)(16,19,26,28,21,27,20,18)(17,23,30,22,25,31,24,29)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 19823);