Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#32697 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2749 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,10,14,9,15,27,35)(6,11,19,31,18,30,28,33)(7,20,23,24,22,12,32,17)(8,13,26,16,25,21,29,34);(4,6)(5,11)(7,21)(8,24)(9,18)(10,28)(12,29)(13,22)(14,33)(15,30)(16,23)(17,25)(19,27)(20,26)(31,35)(32,34);(4,7,9,22)(5,12)(6,16)(8,11,25,30)(10,23,27,32)(13,28)(14,17)(15,20)(18,34)(19,21)(24,35)(26,31,29,33);(4,8,9,25)(5,13,15,21)(6,17,18,24)(7,11,22,30)(10,29,27,26)(12,28,20,19)(14,34,35,16)(23,33,32,31)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25685);