Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C23×C42 |
| direct product | C2×C3×#1747 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C42 |
| commutator | C4 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | #376 |
| prime radical | C24×C3 | #41 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2)(35,36,37);(3,4)(5,13,16,26)(6,10)(7,11)(8,12)(9,15,21,29)(14,25,28,33)(17,22)(18,23)(19,24)(20,27,31,34)(30,32);(3,5)(4,9,12,21)(6,14)(7,15)(8,16)(10,20,23,31)(11,13,24,26)(17,27)(18,28)(19,29)(22,25,32,33)(30,34);(3,6,7,17,8,18,19,30)(4,10,11,22,12,23,24,32)(5,14,15,27,16,28,29,34)(9,20,13,25,21,31,26,33)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13357);