Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7092 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#7098 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | #331 |
| commutator | C4×C8 | C2×C3×C4 |
| Frattini, cosocle | C82 | C22×C3 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C82 ◁ G | |
142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3)(4,5,6,11,7,10,14,18,8,12,15,19,16,9,13,17)(21,25)(22,28)(23,30)(26,35)(27,34)(29,32)(31,33);(5,9)(6,13)(7,16)(10,12)(11,19)(14,15)(20,21,24,27)(22,29,31,35)(23,32,30,26)(25,33,34,28)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 86);