Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18654 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C2×C3×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,9)(6,7)(10,11)(12,13,17,14)(15,22,23,27)(16,21,26,19)(18,25,20,24);(4,5)(6,9)(7,10)(8,11)(12,14,17,13)(15,20,23,18)(16,19,26,21)(22,25,27,24);(4,6,8,7)(5,9,11,10)(12,15)(13,18)(14,20)(16,25)(17,23)(19,27)(21,22)(24,26);(4,6,8,7)(5,10,11,9)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11642);