Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60750 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60864 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C2×C3×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C42 ◁ G | |
2036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(5,9)(6,7)(10,11)(12,13)(14,20)(15,18)(16,19)(17,22)(21,26)(23,25)(24,27);(12,14,16,22)(13,17,19,20)(15,23,24,21)(18,26,27,25);(4,5)(6,9)(7,11)(8,10)(12,15,16,24)(13,18,19,27)(14,21,22,23)(17,25,20,26);(13,19)(17,20)(18,27)(25,26);(4,6,8,7)(5,9,10,11)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53852);