Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #63091 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C214 |
| direct product | C3×#63091 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C24×D4 |
| commutator, Frattini, cosocle | C4 | C26×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C3×#585 ◁ G | |
10894 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)(6,11)(7,12)(8,10)(9,14)(13,20)(15,23)(16,24)(17,21)(18,26)(19,22)(25,30)(27,31)(28,33)(29,32)(34,35);(5,10)(8,20)(11,21)(12,22)(13,14)(17,30)(19,31)(23,32)(24,25)(26,27)(29,35)(33,34);(4,6)(5,11)(7,15)(8,17)(9,16)(10,21)(12,23)(13,25)(14,24)(18,28)(19,29)(20,30)(22,32)(26,33)(27,34)(31,35);(4,7)(5,12)(6,15)(8,19)(9,18)(10,22)(11,23)(13,27)(14,26)(16,28)(17,29)(20,31)(21,32)(24,33)(25,34)(30,35);(7,18)(12,26)(15,28)(19,31)(22,27)(23,33)(29,35)(32,34);(6,16)(11,24)(15,28)(17,30)(21,25)(23,33)(29,35)(32,34)246 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56079);