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×#18651 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3499 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
1126 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)(4,5,8,11)(6,10,7,9)(13,18)(14,22)(15,16)(20,21)(23,24)(25,26);(5,9)(6,7)(10,11)(12,13,17,21)(14,19,23,27)(15,18,16,20)(22,25,24,26);(12,14)(13,19)(15,22)(16,24)(17,23)(18,25)(20,26)(21,27);(4,6,8,7)(5,9,11,10)(12,15,17,16)(13,20,21,18)(14,22,23,24)(19,26,27,25)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11639);