Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18517 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| socle | C22×C3 | C2×#120 |
| prime radical | C3×D4×#41 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
1204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5,7,10)(6,8,11,9)(12,13,17,22)(14,23,26,27)(15,20,16,21)(18,24,19,25);(5,8)(6,11)(9,10)(12,14)(13,18)(15,24)(16,25)(17,26)(19,22)(20,23)(21,27);(12,14)(13,19)(15,24)(16,25)(17,26)(18,22)(20,27)(21,23);(12,15,17,16)(13,20,22,21)(14,24,26,25)(18,23,19,27)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11505);