Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#21784 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2308 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
1278 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)(6,14,17,10)(7,18,20,30)(8,13)(9,12)(11,19,27,22)(15,31)(16,29,21,25)(23,28)(24,32)(26,34)(33,35);(4,6)(5,10)(7,15)(8,21)(9,17)(11,26)(12,14)(13,29)(16,23)(18,35)(19,34)(20,33)(22,32)(24,27)(25,28)(30,31);(4,7,9,20)(5,11,12,27)(6,15,17,33)(8,22,23,19)(10,24,14,26)(13,30,28,18)(16,34,21,32)(25,31,29,35);(4,6)(5,10)(7,15)(8,16)(9,17)(11,24)(12,14)(13,25)(18,35)(19,32)(20,33)(21,23)(22,34)(26,27)(28,29)(30,31)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14772);