Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#99 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C2×Q8 | C24×C3 |
| prime radical | C22×C3×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C24×C3 ◁ C23×C3×Q8 ◁ G | |
1198 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);(3,4,7,11,8,12,23,30)(5,13,16,31,14,27,21,33)(6,17,19,29,20,10,22,28)(9,24,26,18,25,32,34,15);(3,5)(4,9)(6,18)(7,21)(8,14)(10,27)(11,26)(12,25)(13,17)(15,20)(16,23)(19,24)(22,32)(28,33)(29,31)(30,34);(5,14)(9,25)(13,27)(15,18)(16,21)(24,32)(26,34)(31,33)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26354);