Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#40619 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5)(6,11)(7,10)(8,9)(20,21,24,22)(23,26,27,25);(5,9)(7,11)(12,13)(14,18)(15,16)(17,19)(21,22)(25,26);(5,9)(6,10)(13,16)(17,18)(20,22,24,21)(23,26,27,25);(4,6,8,10)(5,7,9,11)(12,13)(14,17)(15,16)(18,19)(20,23,24,27)(21,25,22,26);(4,7)(5,10)(6,9)(8,11)(12,14)(13,17)(15,19)(16,18)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 33607);