Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#20786 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C23×C3 | C23×C4 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,16)(15,17)(18,19)(20,21,24,28,23,27,31,35)(22,25,30,32,29,26,33,34);(5,8)(9,11)(12,14,15,18)(13,16,17,19)(20,22,23,29)(21,25,27,26)(24,30,31,33)(28,32,35,34);(13,17)(16,19)(20,22)(21,26)(23,29)(24,30)(25,27)(28,34)(31,33)(32,35);(4,6,7,10)(5,9,8,11)(22,29)(25,26)(30,33)(32,34)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13774);