Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#27792 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G | |
810 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)(10,12)(11,13)(14,15)(16,17,21,23)(18,24,27,31)(19,28,26,22)(20,30,29,25);(4,6,7,5)(8,10,11,14)(9,12,13,15)(16,18,20,26,21,27,29,19)(17,22,25,24,23,28,30,31);(5,6)(9,13)(12,15)(17,23)(22,28)(24,31)(25,30);(16,19,29,27,21,26,20,18)(17,24,30,22,23,31,25,28)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20780);