Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×#3069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C2×C3×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
2618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2)(35,36,37);(3,4,9,14)(5,15,17,26)(6,20,22,28)(7,13,8,12)(10,19,25,18)(11,23,27,21)(16,30,31,24)(29,33,34,32);(3,5,7,18,9,17,8,19)(4,10,12,15,14,25,13,26)(6,16,21,32,22,31,23,33)(11,24,20,29,27,30,28,34);(3,6)(4,11)(5,16)(7,21)(8,23)(9,22)(10,24)(12,20)(13,28)(14,27)(15,29)(17,31)(18,32)(19,33)(25,30)(26,34);(5,17)(10,25)(15,26)(16,31)(18,19)(24,30)(29,34)(32,33)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53456);