Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#17604 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3494 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G | |
1004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,10,12,23)(7,18)(11,22,26,33)(13,24,14,19)(16,30)(20,32)(25,31,27,34)(28,35);(4,5)(6,11)(7,19)(8,13)(9,14)(10,20)(12,21)(15,25)(16,31)(17,27)(18,23)(22,28)(24,32)(26,29)(30,33)(34,35);(4,6,8,17)(5,11,13,27)(7,16,18,30)(9,15,21,29)(10,22,24,34)(12,26,14,25)(19,31,23,33)(20,28,32,35);(6,15)(11,25)(16,28)(17,29)(22,31)(26,27)(30,35)(33,34)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10592);