Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#23953 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3493 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,19)(15,26)(16,21)(17,30)(18,22)(20,28)(23,39)(24,33)(25,35)(27,37)(29,38)(31,34)(32,40)(36,42)(41,43);(4,6,7,11)(5,8,9,10)(12,14)(13,19)(15,23)(16,24)(17,31)(18,25)(20,36)(21,33)(22,35)(26,39)(27,40)(28,42)(29,41)(30,34)(32,37)(38,43);(12,15,18,29)(13,20,22,37)(14,23,25,41)(16,28,17,27)(19,32,35,36)(21,38,30,26)(24,42,31,40)(33,39,34,43);(5,9)(8,10)(12,16,18,17)(13,21,22,30)(14,24,25,31)(15,27,29,28)(19,33,35,34)(20,26,37,38)(23,40,41,42)(32,43,36,39)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16941);