Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#29316 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C43 ◁ G | |
634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,10)(7,8)(9,11)(20,21,24,28,23,27,32,26)(22,29,31,34,30,25,33,35);(5,8)(9,10)(12,13,15,17)(14,18,19,16)(20,22,23,30)(21,25,27,29)(24,31,32,33)(26,34,28,35);(12,14,15,19)(13,16,17,18)(21,26)(24,32)(25,34)(27,28)(29,35)(31,33);(4,6,7,11)(5,9,8,10)(21,27)(25,29)(26,28)(34,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22304);