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×#63042 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6758 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)(6,14)(7,13)(8,12)(9,11)(10,18)(15,24)(16,22)(17,28)(19,26)(20,27)(21,25)(23,31)(29,34)(30,33)(32,35);(4,6)(5,10)(7,17)(8,18)(9,16)(11,24)(12,14)(13,23)(15,21)(19,30)(20,31)(22,25)(26,34)(27,28)(29,32)(33,35);(5,11)(6,15)(10,14)(12,25)(13,26)(16,18)(17,29)(22,24)(23,28)(27,35)(30,31)(33,34);(5,12)(6,16)(10,22)(11,25)(13,27)(14,24)(15,18)(17,30)(23,33)(26,35)(28,34)(29,31);(4,7,8,20)(5,13,12,27)(6,17,18,31)(9,19,21,32)(10,23,14,28)(11,26,25,35)(15,29,16,30)(22,33,24,34);(7,19)(13,26)(17,30)(20,32)(23,34)(27,35)(28,33)(29,31)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56030);