Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19059 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,10)(6,13)(7,8)(12,15)(14,16)(17,19)(21,25)(23,30)(26,34)(27,31)(29,32)(33,35);(4,5)(6,11)(7,15)(8,10)(9,12)(13,19)(14,18)(16,17)(20,21,24,27)(22,26,28,35)(23,31,30,25)(29,33,32,34);(4,6)(5,11)(7,13)(8,16)(9,14)(10,17)(12,18)(15,19)(20,22)(21,26)(23,32)(24,28)(25,33)(27,35)(29,30)(31,34);(6,14)(11,18)(13,16)(17,19)(22,28)(26,35)(29,32)(33,34)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12047);