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×#1902 |
| 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 | |
1654 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)(5,14)(6,18,21,28)(7,22)(8,12)(9,13)(10,16)(11,20,27,19)(15,30,32,24)(17,25)(23,26)(29,31,34,33);(3,5,7,16,9,17,8,23)(4,10,12,14,13,26,22,25)(6,15,20,33,21,32,19,31)(11,24,28,29,27,30,18,34);(3,6)(4,11)(5,15)(7,20)(8,19)(9,21)(10,24)(12,28)(13,27)(14,29)(16,33)(17,32)(18,22)(23,31)(25,34)(26,30)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13981);