Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31076 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#32702 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | #2782 |
| prime radical | C3×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,9,13,10,15,8,14)(6,16,19,11,20,27,18,28)(7,21,24,12,25,30,23,31)(17,32,33,29,34,26,22,35);(4,6)(5,11)(7,22)(8,18)(9,19)(10,20)(12,29)(13,27)(14,16)(15,28)(17,24)(21,32)(23,34)(25,33)(26,30)(31,35);(4,7,8,23,10,25,9,24)(5,12,13,30,15,31,14,21)(6,17,19,33,20,34,18,22)(11,26,16,29,28,32,27,35);(6,18,20,19)(11,27,28,16)(17,22,34,33)(26,35,32,29)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25690);