Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C214 |
| direct product | C2×C3×#3584 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C26×C3 |
| socle | C22×C3 | C23×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×#41 ◁ G | |
10858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 37 points:
(1,2)(35,36,37);(3,4)(5,14)(6,11)(7,12)(8,21)(9,13)(10,16)(15,28)(17,24)(18,26)(19,27)(20,31)(22,25)(23,30)(29,34)(32,33);(3,5)(4,10)(6,15)(7,16)(8,22)(9,17)(11,23)(12,24)(13,25)(14,21)(18,29)(19,30)(20,32)(26,33)(27,34)(28,31);(3,6)(4,11)(5,15)(7,19)(8,20)(9,18)(10,23)(12,27)(13,26)(14,28)(16,30)(17,29)(21,31)(22,32)(24,34)(25,33);(6,18)(11,26)(15,29)(19,20)(23,33)(27,31)(28,34)(30,32);(3,7,9,8)(4,12,13,21)(5,16,17,22)(6,19,18,20)(10,24,25,14)(11,27,26,31)(15,30,29,32)(23,34,33,28)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56075);