Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#9297 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C23×C3 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
454 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)(4,5,8,12,9,11,7,17)(6,13,16,18,15,19,14,10)(20,21,24,26,25,29,23,28)(22,30,32,27,33,34,31,35);(4,6)(5,10,11,18)(7,14)(8,16)(9,15)(12,13,17,19)(21,26)(23,24)(27,30)(28,29)(31,32)(34,35);(5,11)(7,8)(10,18)(14,16)(20,22,25,33)(21,27,29,35)(23,32,24,31)(26,30,28,34)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2285);