Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10016 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | #392 |
| prime radical | C22×C3×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
560 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)(4,5,8,11)(6,12,14,19)(7,15,17,10)(9,16,18,13)(20,21,25,29)(22,30,33,34)(23,28,24,27)(26,32,35,31);(4,6,8,14)(5,9,11,18)(7,16,17,13)(10,19,15,12)(20,22,25,33)(21,26,29,35)(23,32,24,31)(27,34,28,30);(9,18)(12,19)(21,27)(23,24)(26,34)(28,29)(30,35)(31,32)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3004);