Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9996 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10003 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C23×C3 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
500 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)(4,5,9,12)(6,13,15,18)(7,11,8,16)(10,17,19,14)(20,21)(22,28)(23,31)(24,27)(25,30)(26,33)(29,35)(32,34);(4,6,9,15)(5,10,12,19)(7,14,8,17)(11,18,16,13)(25,34)(28,35);(4,7,9,8)(5,11,12,16)(6,14,15,17)(10,18,19,13)(20,22)(21,25)(23,32)(24,29)(26,35)(27,34)(28,31)(30,33)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2991);