Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3×C4 · Schur multiplier (H2) = C214 |
| direct product | C2×C3×#3573 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C24 |
| commutator | C2 | C25×C3×C4 |
| Frattini, cosocle | C4 | C26×C3 |
| socle | C22×C3 | C24×C4 |
| prime radical | C2×C3×#585 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ 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);(3,4,7,11,8,12,18,24)(5,9,15,21,14,20,27,31)(6,10,17,23,16,22,28,32)(13,19,26,30,25,29,33,34);(3,5)(4,9)(6,13)(7,15)(8,14)(10,19)(11,21)(12,20)(16,25)(17,26)(18,27)(22,29)(23,30)(24,31)(28,33)(32,34);(3,6)(4,10)(5,13)(7,17)(8,16)(9,19)(11,23)(12,22)(14,25)(15,26)(18,28)(20,29)(21,30)(24,32)(27,33)(31,34);(6,16)(10,22)(13,25)(17,28)(19,29)(23,32)(26,33)(30,34);(5,14)(9,20)(13,25)(15,27)(19,29)(21,31)(26,33)(30,34)408 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56063);