Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C16 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#1400 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C22⋊C4 |
| commutator | C22 | C22×C3×C16 |
| Frattini, cosocle | C2×C16 | C23×C3 |
| socle | C22×C3 | C22⋊C16 |
| prime radical | C24×C3 | C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C22×C3×C16 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2)(35,36,37);(3,4,5,10,6,11,14,22,7,12,15,23,16,24,27,32,8,9,13,19,17,20,25,29,18,21,26,30,28,31,33,34);(4,9)(5,13)(11,20)(12,21)(14,25)(15,26)(24,31)(27,33)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6543);