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×#36196 |
| 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 | |
1694 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)(4,5,8,10)(6,7,9,11)(12,13,16,19)(14,17,15,18)(20,21,23,25)(22,24,26,27);(4,6)(5,7)(8,9)(10,11)(12,13)(14,17)(15,18)(16,19)(20,21)(22,24)(23,25)(26,27);(4,7)(5,9)(6,10)(8,11)(12,13)(14,17)(15,18)(16,19);(4,6)(5,7)(8,9)(10,11)(12,14,16,15)(13,17,19,18);(4,7)(5,9)(6,10)(8,11)(12,15,16,14)(13,18,19,17)(20,22,23,26)(21,24,25,27)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29184);