Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60685 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×#117 ◁ G | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,19)(15,27)(16,21)(17,32)(18,22)(20,30)(23,38)(24,34)(25,35)(26,41)(28,42)(29,37)(31,43)(33,40)(36,39);(5,8)(9,11)(12,14)(13,19)(15,28)(16,25)(17,26)(18,24)(20,36)(21,35)(22,34)(23,31)(27,42)(29,40)(30,39)(32,41)(33,37)(38,43);(12,15,17,30,18,31,16,29)(13,20,21,27,22,37,32,43)(14,23,26,40,24,28,25,39)(19,33,35,38,34,36,41,42);(4,6,7,10)(5,9,8,11)(14,24)(19,34)(23,28)(25,26)(33,36)(35,41)(38,42)(39,40);(12,16,18,17)(13,21,22,32)(14,25,24,26)(15,29,31,30)(19,35,34,41)(20,27,37,43)(23,39,28,40)(33,38,36,42)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53673);