Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#25825 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C23×C3 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
934 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)(4,5,8,11)(6,10,7,9)(12,13,17,21)(14,22,25,19)(15,18,20,16)(23,27,26,24);(4,6,8,7)(5,9,11,10)(12,14)(13,19)(15,26)(16,24)(17,25)(18,27)(20,23)(21,22);(12,15,18,21,17,20,16,13)(14,23,24,22,25,26,27,19);(4,7,8,6)(5,10,11,9)(12,16,17,18)(13,20,21,15)(14,24,25,27)(19,23,22,26)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18813);