Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#1905 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2)(35,36,37);(3,4,8,12,9,14,7,13)(5,15,18,25,19,28,17,10)(6,20,22,11,23,26,21,27)(16,24,31,33,32,34,30,29);(3,5)(4,10,14,25)(6,16)(7,17)(8,18)(9,19)(11,24,27,34)(12,15,13,28)(20,29,26,33)(21,30)(22,31)(23,32);(3,6,9,23)(4,11,14,27)(5,16,19,32)(7,22,8,21)(10,24,25,34)(12,20,13,26)(15,29,28,33)(17,31,18,30)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13985);