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×#63046 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 35 points:
(1,2,3)(4,5,9,13)(6,15,17,11)(7,18,21,28)(8,12,23,27)(10,14,24,19)(16,29,31,25)(20,30,33,35)(22,26,34,32);(4,6,9,17)(5,11,13,15)(7,19,21,14)(8,16,23,31)(10,18,24,28)(12,25,27,29)(20,32,33,26)(22,30,34,35);(4,7)(5,6)(8,20)(9,21)(10,11)(12,16)(13,17)(14,18)(15,24)(19,28)(22,25)(23,33)(26,30)(27,31)(29,34)(32,35);(4,5)(6,7)(8,12)(9,13)(10,14)(11,18)(15,28)(16,20)(17,21)(19,24)(22,26)(23,27)(25,30)(29,35)(31,33)(32,34);(4,8,9,23)(5,12,13,27)(6,16,17,31)(7,20,21,33)(10,22,24,34)(11,25,15,29)(14,26,19,32)(18,30,28,35);(8,22)(12,26)(16,30)(20,25)(23,34)(27,32)(29,33)(31,35)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56034);