Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#9790 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #390 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C22×C3×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
384 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)(4,5,8,12,9,13,7,10)(6,14,17,19,15,18,16,11)(20,21,23,27,24,26,31,32)(22,25,29,34,28,33,30,35);(5,10,13,12)(6,15)(11,14,19,18)(16,17)(20,22,23,30,24,28,31,29)(21,25,32,34,26,33,27,35);(4,6,7,16,9,15,8,17)(5,11,12,14,13,19,10,18)(22,28)(25,33)(29,30)(34,35)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2778);