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×#32699 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2749 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
538 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)(4,5,10,15,9,14,22,30)(6,7,12,19,17,11,23,26)(8,13,21,29,20,28,24,31)(16,25,32,35,33,18,27,34);(4,6)(5,11)(7,14)(8,16)(9,17)(10,23)(12,22)(13,18)(15,19)(20,33)(21,27)(24,32)(25,28)(26,30)(29,35)(31,34);(4,7)(5,12,14,23)(6,15,17,30)(8,18)(9,11)(10,19)(13,27,28,32)(16,31,33,29)(20,25)(21,34)(22,26)(24,35);(4,8,9,20)(5,13,14,28)(6,16,17,33)(7,18,11,25)(10,24,22,21)(12,27,23,32)(15,31,30,29)(19,35,26,34)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25687);