Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#12666 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C42 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C22×C3 | C2×#97 |
| prime radical | C23×C3 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5,6,7)(8,9,12,17,11,16,20,23,13,18,21,10,15,14,22,19);(5,7)(9,14,18,16)(10,19,17,23)(12,21)(20,22);(8,10,20,9,15,23,12,14,13,17,22,18,11,19,21,16)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5654);