Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×#3290 |
| 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 | |
2650 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)(5,14)(6,11)(7,24)(8,13)(9,17)(10,22)(12,25)(15,27)(16,33)(18,29)(19,28)(20,26)(21,31)(23,30)(32,34);(3,5)(4,9)(6,20)(7,18)(8,19)(10,27)(11,32)(12,28)(13,29)(14,24)(15,23)(16,21)(17,25)(22,34)(26,31)(30,33);(3,6)(4,10)(5,15)(7,21)(8,23)(9,26)(11,24)(12,30)(13,31)(14,33)(16,17)(18,34)(19,20)(22,25)(27,29)(28,32);(3,6)(4,11)(5,16)(7,22)(8,23)(9,27)(10,24)(12,31)(13,30)(14,33)(15,17)(18,20)(19,34)(21,25)(26,29)(28,32)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54089);