Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×#2916 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C4 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C22×C4⋊C4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×#41 ◁ G | |
2586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2)(35,36,37);(3,4,8,12,9,13,7,22)(5,14,17,24,18,28,16,10)(6,11,21,26,19,25,20,32)(15,27,31,33,29,34,30,23);(3,5,7,16,9,18,8,17)(4,10,12,14,13,24,22,28)(6,15,20,30,19,29,21,31)(11,23,26,27,25,33,32,34);(3,6)(4,11)(5,15)(7,20)(8,21)(9,19)(10,23)(12,26)(13,25)(14,27)(16,30)(17,31)(18,29)(22,32)(24,33)(28,34);(6,19)(11,25)(15,29)(20,21)(23,33)(26,32)(27,34)(30,31)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53100);