Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#21140 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×#97 ◁ G | |
764 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)(6,9)(7,8)(10,11)(12,13,17,22,18,21,16,30)(14,19,26,33,24,34,25,40)(15,27,28,20,29,36,31,35)(23,37,39,32,38,42,41,43);(5,8)(9,11)(12,14,17,26,18,24,16,25)(13,19,22,33,21,34,30,40)(15,23,28,39,29,38,31,41)(20,32,36,42,35,43,27,37);(12,15)(13,20)(14,23)(16,31)(17,28)(18,29)(19,32)(21,35)(22,36)(24,38)(25,41)(26,39)(27,30)(33,42)(34,43)(37,40);(4,6,7,10)(5,9,8,11)(13,21)(14,24)(20,35)(22,30)(23,38)(25,26)(27,36)(39,41)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14128);