Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#2080 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | #397 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2)(35,36,37);(3,4)(5,13,16,26)(6,10)(7,11)(8,12)(9,15,23,20)(14,29,17,28)(18,24)(19,25)(21,27)(22,31,33,30)(32,34);(3,5)(4,9)(6,17)(7,20)(8,16)(10,22)(11,26)(12,23)(13,27)(14,19)(15,21)(18,30)(24,29)(25,33)(28,34)(31,32);(3,6,7,18,8,19,21,32)(4,10,11,24,12,25,27,34)(5,14,15,30,16,17,20,31)(9,22,13,28,23,33,26,29)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16636);