Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#9845 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #127 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | #390 |
| prime radical | C23×C3 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
478 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)(4,5,7,10,8,11,6,9)(12,13)(14,20)(15,17)(16,24)(18,23)(19,21)(22,26)(25,27);(5,9,11,10)(13,17)(18,25)(19,24)(20,26);(12,14,21,27,15,22,16,23)(13,18,24,26,17,25,19,20)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2833);