Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C3×#1835 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C2×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C22×C3×C42 ◁ G | |
1006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2)(31,32,33);(3,4,5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,25)(18,21)(19,29)(20,24)(22,27)(23,26)(28,30);(8,11)(12,14)(15,17,26,28,20,27,19,18)(16,21,29,22,24,30,23,25);(7,9,10,13)(8,12,11,14)(15,18,19,27,20,28,26,17)(16,22,23,21,24,25,29,30)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13747);