Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#11029 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #121 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C24×C3 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
1242 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)(6,7)(12,13,17,23,18,21,16,22)(14,24,27,35,28,19,26,36)(15,29,32,38,33,20,31,39)(25,34,41,43,42,40,30,37);(5,7)(8,9)(10,11)(12,14,18,28)(13,19,21,24)(15,30,33,41)(16,27,17,26)(20,37,29,43)(22,36,23,35)(25,32,42,31)(34,39,40,38);(9,11)(12,15,18,33)(13,20,21,29)(14,25,28,42)(16,32,17,31)(19,34,24,40)(22,39,23,38)(26,41,27,30)(35,43,36,37)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4017);