Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#32656 |
| 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 ◁ C2×C3×C4×Q8 ◁ G | |
664 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)(4,5,8,10)(6,7,11,9)(12,13)(14,18)(15,17)(16,19)(20,21,24,27)(22,25,29,33)(23,26,30,34)(28,32,31,35);(4,6,8,11)(5,9,10,7)(12,14)(13,16)(15,19)(17,18)(22,28)(23,30)(25,32)(26,34)(29,31)(33,35);(13,17)(16,18)(20,22,24,29)(21,25,27,33)(23,31,30,28)(26,35,34,32);(4,7,8,9)(5,6,10,11)(14,19)(16,18)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25644);