Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10013 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | #392 |
| prime radical | C3×#3100 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,8)(6,10)(9,11)(12,13)(14,20)(15,23)(16,19)(17,21)(18,25)(22,27)(24,26);(4,5,7,9)(6,11,10,8)(12,14,16,22)(13,17,19,26)(15,24,25,21)(18,27,23,20);(4,5)(6,11)(7,9)(8,10)(17,26)(20,27)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3001);