Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#63044 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 43 points:
(1,2,3)(4,5)(6,10)(7,8)(9,11)(13,18)(19,29)(20,30)(21,31)(32,40)(33,41)(34,35)(42,43);(5,8)(9,10)(13,18)(14,22)(20,30)(21,31)(23,36)(24,37)(34,35)(38,39);(4,5)(6,9)(7,8)(10,11)(12,13)(14,19)(15,20)(16,21)(17,18)(22,29)(23,32)(24,33)(25,34)(26,30)(27,35)(28,31)(36,40)(37,41)(38,42)(39,43);(4,6)(5,9)(7,11)(8,10)(12,14)(13,19)(15,23)(16,24)(17,22)(18,29)(20,32)(21,33)(25,38)(26,36)(27,39)(28,37)(30,40)(31,41)(34,42)(35,43);(12,15,17,26)(13,20,18,30)(14,23,22,36)(16,27,28,25)(19,32,29,40)(21,35,31,34)(24,39,37,38)(33,43,41,42);(12,16,17,28)(13,21,18,31)(14,24,22,37)(15,25,26,27)(19,33,29,41)(20,34,30,35)(23,38,36,39)(32,42,40,43)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56032);