Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×#3289 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
2620 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);(3,4,9,15)(5,16,20,31)(6,12,22,30)(7,14,8,13)(10,23,27,19)(11,24,28,21)(17,25,34,26)(18,32,33,29);(3,5,9,20)(4,10,15,27)(6,17,22,34)(7,23,8,19)(11,25,28,26)(12,29,30,32)(13,31,14,16)(18,24,33,21);(3,6)(4,11)(5,17)(7,21)(8,24)(9,22)(10,25)(12,13)(14,30)(15,28)(16,32)(18,23)(19,33)(20,34)(26,27)(29,31);(3,6)(4,12)(5,18)(7,24)(8,21)(9,22)(10,26)(11,13)(14,28)(15,30)(16,32)(17,23)(19,34)(20,33)(25,27)(29,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54087);