Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#17679 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3514 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×#41 ◁ G | |
942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,7,11,8,9,14,16)(6,10,13,17,12,15,18,19)(20,21)(22,28,30,35)(23,27)(24,26)(25,29,33,31)(32,34);(5,9)(7,14)(10,15)(13,18)(20,22)(21,25)(23,31)(24,30)(26,33)(27,35)(28,34)(29,32);(4,6)(5,10)(7,13)(8,12)(9,15)(11,17)(14,18)(16,19)(21,26)(25,33)(27,34)(28,35);(6,12)(10,15)(13,18)(17,19)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10667);