Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#13228 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#99 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×Q8 | C23×C3 |
| prime radical | C22×C3×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×Q8 ◁ G | |
490 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)(5,7)(8,9,13,17,14,18,11,15)(10,19,21,12,16,20,23,22);(8,10,20,17,14,16,19,15)(9,13,23,12,18,11,21,22);(4,5,6,7)(8,11,14,13)(9,15,18,17)(10,21,16,23)(12,20,22,19)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6216);