Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#17609 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G | |
944 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)(5,10,12,24)(7,18)(11,23,27,34)(13,25,14,19)(16,31)(20,22)(26,32,28,35)(29,33);(4,5,8,13)(6,11,17,28)(7,19,18,24)(9,14,21,12)(10,22,25,20)(15,26,30,27)(16,32,31,34)(23,33,35,29);(4,6,8,17)(5,11,13,28)(7,16,18,31)(9,15,21,30)(10,23,25,35)(12,27,14,26)(19,32,24,34)(20,29,22,33);(6,15)(11,26)(16,29)(17,30)(23,32)(27,28)(31,33)(34,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10597);