Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#8189 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C42 | C2×D4 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | C4×C22⋊C4 |
| prime radical | C2×C3×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,8,12,9,13,7,10)(6,11,15,18,16,19,14,17)(20,21)(22,25)(23,27)(24,26)(28,32)(29,31)(30,33)(34,35);(5,10,13,12)(11,17,19,18)(20,22,23,29,24,28,30,34)(21,25,27,31,26,32,33,35);(4,6,8,15,9,16,7,14)(5,11,12,18,13,19,10,17)(21,26)(22,28)(27,33)(29,34)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1177);