Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#22427 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#546 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G | |
1606 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)(5,7)(8,9,10,11)(12,13)(14,17)(15,18)(16,19)(20,23)(21,25)(22,26)(24,27);(4,5)(6,7)(12,14)(13,17)(15,22)(16,21)(18,26)(19,25)(20,24)(23,27);(13,18)(14,21)(15,23)(17,24)(19,20)(25,26);(5,7)(13,19)(17,25)(18,20)(24,26)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 15415);