Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#22513 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4⋊C4 ◁ G | |
1114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,8)(9,10)(12,13)(14,17)(15,22,23,18)(16,19)(20,26,27,24)(21,25);(4,5,7,8)(6,10,11,9)(12,14,16,21)(13,17,19,25)(15,20,23,27)(18,24,22,26);(12,15)(13,18)(14,20)(16,23)(17,24)(19,22)(21,27)(25,26);(4,6,7,11)(5,9,8,10)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 15501);