Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#1791 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #120 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | #379 |
| prime radical | C2×C3×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
3662 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);(3,4,7,12,8,11,20,25)(5,13,16,26,15,9,19,22)(6,10,18,23,17,24,30,34)(14,21,29,32,28,27,31,33);(3,5)(4,9)(6,14)(7,19)(8,15)(10,21)(11,13)(12,26)(16,20)(17,28)(18,31)(22,25)(23,33)(24,27)(29,30)(32,34);(3,6)(4,10)(5,14)(7,18)(8,17)(9,21)(11,24)(12,23)(13,27)(15,28)(16,29)(19,31)(20,30)(22,32)(25,34)(26,33)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13599);