Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C82 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#7141 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C8 | D4 |
| commutator | C4 | C3×C82 |
| Frattini, cosocle | C2×C4×C8 | C22×C3 |
| socle | C22×C3 | #336 |
| prime radical | C23×C3 | #97 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C8 ◁ C2×C3×C4×C8 ◁ G | |
254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3)(12,13,16,21,17,22,31,39,18,23,32,40,15,20,29,37)(14,24,26,41,25,35,42,38,27,36,30,43,28,19,33,34);(4,5,6,8,7,9,10,11)(12,14)(13,19)(15,28)(16,30)(17,25)(18,27)(20,36)(21,38)(22,24)(23,35)(26,32)(29,42)(31,33)(34,40)(37,41)(39,43)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 129);