Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#29652 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
940 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)(4,5,8,11)(6,12,14,9)(7,10,15,17)(13,18,19,16)(20,21)(22,28)(23,31)(24,27)(25,29)(26,32)(30,35)(33,34);(4,6,8,14)(5,9,11,12)(7,13,15,19)(10,16,17,18)(23,32)(25,34)(26,31)(29,33);(4,7)(5,10)(6,13)(8,15)(9,16)(11,17)(12,18)(14,19)(20,22,24,30)(21,25,27,34)(23,33,32,29)(26,35,31,28);(7,15)(10,17)(13,19)(16,18)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22640);