Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#27734 |
| 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×C4×C4⋊C4 ◁ G | |
606 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)(12,13)(14,16)(15,17)(18,19)(20,21,25,31,26,29,24,30)(22,27,33,35,23,34,32,28);(5,8)(9,10)(12,14,15,18)(13,16,17,19)(20,22,24,32,26,23,25,33)(21,27,30,28,29,34,31,35);(13,17)(16,19)(20,23,26,22)(21,28,29,35)(24,32,25,33)(27,31,34,30);(4,6,7,11)(5,9,8,10)(21,29)(27,34)(28,35)(30,31)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20722);