Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#22462 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | D42 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | D4×C22⋊C4 |
| prime radical | C3×#3584 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
1320 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)(6,15,19,17)(7,20)(8,23,9,13)(11,24,27,18)(12,25)(16,31)(21,29)(22,28)(26,35)(30,33)(32,34);(4,6)(5,11)(7,16)(8,24)(9,18)(10,19)(12,26)(13,17)(14,27)(15,23)(20,30)(21,34)(22,35)(25,33)(28,32)(29,31);(4,7)(5,12)(6,16)(8,21)(9,25)(10,22)(11,26)(13,28)(14,29)(15,30)(17,32)(18,33)(19,35)(20,23)(24,34)(27,31);(4,5)(6,17)(7,12)(8,23)(9,13)(10,14)(11,24)(15,19)(16,32)(18,27)(20,21)(22,29)(25,28)(26,34)(30,35)(31,33)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 15450);