Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#13274 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D8 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C22×C3 | C2×#102 |
| prime radical | C23×C3 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3)(4,5)(6,10)(7,8)(9,11)(12,13,16,21,17,22,31,39,18,23,32,40,30,38,15,20)(14,24,26,36,27,37,25,41,28,42,29,19,33,34,43,35);(5,8)(9,10)(12,14,17,33,18,28,30,27)(13,19,22,41,23,36,38,35)(15,29,16,25,31,26,32,43)(20,37,21,24,39,34,40,42);(4,6,7,11)(5,9,8,10)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6262);