Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33735 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C22×C3 | C22×D8 |
| prime radical | C3×#2215 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,19)(15,25)(16,30)(17,32)(18,22)(20,27)(21,26)(23,36)(24,35)(28,42)(29,39)(31,43)(33,37)(34,41)(38,40);(5,8)(9,11)(12,14,18,17)(13,19,22,32)(15,23,29,28)(16,24,31,26)(20,33,34,38)(21,30,35,43)(25,36,39,42)(27,37,41,40);(12,15,26,40,17,28,31,41,18,29,24,37,14,23,16,27)(13,20,30,36,19,33,35,39,22,34,43,42,32,38,21,25);(4,6,7,10)(5,9,8,11)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26723);