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) = C25×C4 |
| direct product | C2×C3×#2129 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#99 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C2×D4 | C24×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×#41 ◁ G | |
1230 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)(5,13,16,28)(6,10)(7,20,22,11)(8,12)(9,15)(14,29,17,31)(18,33,34,26)(19,27)(21,24)(23,30)(25,32);(3,5,8,16)(4,9,12,24)(6,17,19,14)(7,21,22,15)(10,25,27,23)(11,28,20,13)(18,30,34,32)(26,29,33,31);(3,6,8,19)(4,10,12,27)(5,14,16,17)(7,18,22,34)(9,23,24,25)(11,26,20,33)(13,29,28,31)(15,30,21,32)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26345);