Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#21950 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C22×C4⋊C4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×#41 ◁ G | |
894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13,17,21,18,22,16,31)(14,19,26,34,24,33,25,40)(15,27,29,35,30,41,28,20)(23,36,39,42,37,43,38,32);(4,6)(5,8)(7,11)(9,10)(12,14)(13,19)(15,23)(16,25)(17,26)(18,24)(20,32)(21,34)(22,33)(27,36)(28,38)(29,39)(30,37)(31,40)(35,42)(41,43);(6,11)(8,10)(12,15,16,28,18,30,17,29)(13,20,21,27,22,35,31,41)(14,23,25,38,24,37,26,39)(19,32,34,36,33,42,40,43);(5,9)(8,10)(14,24)(19,33)(23,37)(25,26)(32,42)(34,40)(36,43)(38,39)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14938);