Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C24 |
| direct product | C3×#12982 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | #120 |
| commutator | C2×C4 | C22×C3×C8 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C2×C3 | #1869 |
| prime radical | C3×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C22×C3×C8 ◁ G | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,9,15,10,16,28,34,11,14,29,33,8,13,18,31)(6,17,21,26,19,32,35,7,20,23,27,22,25,12,30,24);(4,6,18,35,8,25,29,21,11,20,28,30,10,19,9,27)(5,12,31,26,13,23,33,24,14,32,34,22,16,17,15,7);(4,7)(5,6)(8,26)(9,17)(10,22)(11,24)(12,18)(13,25)(14,20)(15,27)(16,19)(21,33)(23,29)(28,32)(30,34)(31,35)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5970);