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×#29655 |
| 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 | |
906 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)(22,27)(23,32)(24,29)(25,34)(26,30)(28,33)(31,35);(4,6,8,14)(5,9,11,12)(7,13,15,19)(10,16,17,18)(20,22)(21,27)(23,26)(24,31)(25,33)(28,34)(29,35)(30,32);(4,7)(5,10)(6,13)(8,15)(9,16)(11,17)(12,18)(14,19)(20,23,26,22)(21,28,30,35)(24,33,25,31)(27,34,32,29);(7,15)(10,17)(13,19)(16,18)(20,24,26,25)(21,29,30,34)(22,31,23,33)(27,35,32,28)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22643);