Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10407 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #120 |
| commutator, Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
576 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)(5,6)(8,9)(10,12)(11,13)(14,15)(16,17,19,23)(18,20,22,21);(4,5)(6,7)(8,10)(9,12)(11,14)(13,15)(16,18,19,22)(17,21,23,20);(4,6,7,5)(12,15)(17,22)(18,23)(20,21)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3395);