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×#36256 |
| 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 | |
1838 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)(6,9)(7,8)(10,11)(21,26)(22,27);(4,5)(6,10)(7,8)(9,11)(12,13,15,17)(14,16,19,18)(20,21)(22,23)(24,27)(25,26);(5,8)(9,10)(12,13)(14,18)(15,17)(16,19)(20,22,25,27)(21,23,26,24);(4,5)(6,10)(7,8)(9,11)(12,14)(13,16)(15,19)(17,18)(20,23,25,24)(21,22,26,27);(4,6)(5,9)(7,11)(8,10)(12,13)(14,18)(15,17)(16,19)(20,24,25,23)(21,27,26,22)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29244);