Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#18433 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(7,14)(9,15)(10,17)(12,18)(20,21,26,23)(22,30,33,35)(24,29,25,28)(27,34,31,32);(4,5)(6,9)(7,10)(8,11)(12,16)(13,15)(14,17)(18,19)(20,22,26,33)(21,27,23,31)(24,32,25,34)(28,30,29,35);(4,6)(5,9)(7,12)(8,13)(10,16)(11,15)(14,18)(17,19)(20,23,26,21)(22,31,33,27)(24,28,25,29)(30,34,35,32);(4,7,8,14)(5,10,11,17)(6,12,13,18)(9,16,15,19)(20,24,26,25)(21,28,23,29)(22,32,33,34)(27,30,31,35)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11421);