Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18138 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C25×C3 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,9,13)(6,14,16,30)(7,18,20,32)(8,11,22,26)(10,23,25,15)(12,27,29,21)(17,28,31,35)(19,24,33,34);(4,6,8,17)(5,10,11,24)(7,19,21,15)(9,16,22,31)(12,28,18,14)(13,25,26,34)(20,33,27,23)(29,35,32,30);(7,20)(12,29)(14,30)(15,23)(18,32)(19,33)(21,27)(28,35);(5,11)(10,24)(12,18)(13,26)(14,28)(25,34)(29,32)(30,35)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11126);