Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#29134 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#376 ◁ G | |
690 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)(6,10)(7,8)(9,11)(12,13,15,17)(14,16,18,19)(20,21,25,27,24,29,23,28)(22,26,32,35,30,34,31,33);(5,8)(9,10)(12,14,15,18)(13,16,17,19)(20,22)(21,26)(23,31)(24,30)(25,32)(27,35)(28,33)(29,34);(14,18)(16,19)(21,27)(22,30)(23,25)(26,33)(28,29)(34,35);(4,6,7,11)(5,9,8,10)(22,30)(26,34)(31,32)(33,35)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22122);