Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#12782 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #1790 |
| commutator | C2×Q8 | C22×C3×C4 |
| Frattini, cosocle | C22×Q8 | C23×C3 |
| prime radical | C3×#3544 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C23×C3 ◁ C22×C3×Q8 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,10,16,11,17,9,15)(6,18,20,34,21,32,26,28)(7,22,24,13,8,14,23,31)(12,27,29,19,30,35,33,25);(4,6)(5,12)(7,19)(8,25)(9,26)(10,20)(11,21)(13,28)(14,32)(15,29)(16,33)(17,30)(18,22)(23,27)(24,35)(31,34);(4,7,11,8)(5,13,17,31)(6,19,21,25)(9,24,10,23)(12,28,30,34)(14,16,22,15)(18,29,32,33)(20,27,26,35)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5770);