Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C2×C42 |
| direct product | C3×#15041 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,8)(6,10)(12,13,16,14)(15,18,19,17)(20,21,23,25)(22,26,27,24)(29,32)(30,34);(5,8)(9,11)(13,14)(17,18)(28,29,31,32)(30,33,34,35);(4,5,7,8)(6,9,10,11)(12,14,16,13)(15,18,19,17)(20,22,23,27)(21,24,25,26)(28,29)(30,33)(31,32)(34,35);(4,6)(5,9)(7,10)(8,11)(12,15,16,19)(13,17,14,18)(20,21)(22,24)(23,25)(26,27)(28,30,31,34)(29,33,32,35)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8029);