Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19188 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
714 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)(6,8)(7,10)(9,11)(12,13,17,21)(14,22,26,36)(15,19,27,33)(16,29,23,20)(18,30,32,25)(24,35,38,43)(28,40,37,34)(31,41,42,39);(12,14,23,30,17,26,16,25)(13,18,29,36,21,32,20,22)(15,24,37,41,27,38,28,39)(19,31,40,43,33,42,34,35);(5,8)(6,11)(9,10)(12,15)(13,19)(14,24)(16,28)(17,27)(18,31)(20,34)(21,33)(22,35)(23,37)(25,39)(26,38)(29,40)(30,41)(32,42)(36,43);(15,27)(19,33)(24,38)(28,37)(31,42)(34,40)(35,43)(39,41)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12176);