Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18660 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
908 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)(6,12)(7,15)(8,10)(9,14)(11,17)(13,19)(16,18)(21,26)(22,30)(23,24)(28,29)(31,32)(33,34);(4,6,8,13)(5,9,10,18)(7,16,17,14)(11,19,15,12)(20,21,25,29)(22,27,31,35)(23,26,24,28)(30,33,32,34);(20,22)(21,27)(23,30)(24,32)(25,31)(26,33)(28,34)(29,35);(5,10)(6,13)(11,15)(14,16)(20,23,25,24)(21,28,29,26)(22,30,31,32)(27,34,35,33)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11648);