Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×#12005 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C2×D4 |
| commutator | C2×C4 | C22×C3×C8 |
| Frattini, cosocle | C4×C8 | C23×C3 |
| socle | C22×C3 | C2×C4⋊C8 |
| prime radical | C23×C3 | C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C2×C3×C4×C8 ◁ G | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3)(5,8)(6,9)(10,11)(12,13,15,19,16,20,25,31,17,21,26,32,27,33,37,41)(14,18,23,29,24,30,36,40,22,28,34,38,35,39,42,43);(4,5,9,11,7,10,6,8)(12,14)(13,18)(15,23)(16,24)(17,22)(19,29)(20,30)(21,28)(25,36)(26,34)(27,35)(31,40)(32,38)(33,39)(37,42)(41,43);(14,22)(18,28)(23,34)(24,35)(29,38)(30,39)(36,42)(40,43)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4993);