Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#23406 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
2022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(8,9)(10,18)(11,20)(12,15)(13,16)(14,17)(19,22)(21,23);(4,5)(6,7)(8,10)(9,11)(12,19)(13,20)(14,21)(15,18)(16,23)(17,22);(5,7)(8,11)(9,10)(12,18)(13,23)(14,22)(15,19)(16,20)(17,21);(8,12,14,13)(9,15,17,16)(10,19,21,20)(11,18,22,23)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16394);