Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#36189 |
| 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 | |
1918 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)(6,8)(7,9)(10,11)(12,13,16,18)(14,15,17,19)(20,21)(22,27)(23,26)(24,25);(4,5)(6,10)(7,9)(8,11)(12,14)(13,15)(16,17)(18,19)(21,25)(22,26);(4,5,7,9)(6,8,11,10)(12,15)(13,17)(14,18)(16,19)(20,22,24,26)(21,23,25,27);(4,5)(6,10)(7,9)(8,11)(12,14)(13,15)(16,17)(18,19)(20,23,24,27)(21,26,25,22);(4,6)(5,8)(7,11)(9,10)(12,15)(13,17)(14,18)(16,19)(20,21)(22,23)(24,25)(26,27)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29177);