Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10198 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C3×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,9,13)(6,14,16,19)(7,12,8,11)(10,17,18,15)(20,21,25,29)(22,30,33,34)(23,28,24,27)(26,32,35,31);(4,6,9,16)(5,10,13,18)(7,17,8,15)(11,19,12,14)(20,22,25,33)(21,26,29,35)(23,32,24,31)(27,34,28,30);(4,6)(5,10)(7,15)(8,17)(9,16)(11,14)(12,19)(13,18)(22,31)(23,24)(26,30)(27,28)(32,33)(34,35)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3186);