Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#2155 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #106 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C22×C3 | C2×#106 |
| prime radical | C24×C3 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
658 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)(5,14,17,28)(6,11)(7,20)(8,23)(9,13)(10,16,25,30)(12,21)(15,27,18,32)(19,26)(22,33)(24,29,34,31);(3,5,7,16,8,18,21,31,9,17,22,30,6,15,19,29)(4,10,12,14,11,24,20,27,13,25,26,28,23,34,33,32);(3,6,9,8)(4,11,13,23)(5,15,17,18)(7,19,22,21)(10,24,25,34)(12,20,26,33)(14,27,28,32)(16,29,30,31)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26391);