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) = C24 |
| direct product | C3×#21372 |
| 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 | C24×C3 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×#117 ◁ G | |
714 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)(12,13,17,21,18,22,16,30)(14,23,25,19,26,33,31,34)(15,20,29,36,27,35,28,40)(24,37,39,32,38,43,41,42);(12,14,17,25,18,26,16,31)(13,19,21,33,22,34,30,23)(15,24,29,39,27,38,28,41)(20,32,36,43,35,42,40,37);(5,8)(9,11)(12,15)(13,20)(14,24)(16,28)(17,29)(18,27)(19,32)(21,36)(22,35)(23,37)(25,39)(26,38)(30,40)(31,41)(33,43)(34,42);(4,6,7,10)(5,9,8,11)(15,27)(20,35)(24,38)(28,29)(32,42)(36,40)(37,43)(39,41)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14360);