Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#33362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×#99 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C2×D4 | C24×C3 |
| prime radical | C3×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×Q8 ◁ C2×C3×2−1+4 ◁ G | |
1220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,16)(7,12)(8,13)(9,14,23,25)(10,15)(11,18,26,22)(17,27)(19,28,33,35)(20,29)(21,30)(24,31,34,32);(4,6,10,8)(5,11,15,26)(7,17,20,21)(9,22,23,18)(12,24,29,34)(13,25,16,14)(19,32,33,31)(27,28,30,35);(4,7,10,20)(5,12,15,29)(6,17,8,21)(9,19,23,33)(11,24,26,34)(13,30,16,27)(14,28,25,35)(18,31,22,32);(4,8,10,6)(5,13,15,16)(7,17,20,21)(9,22,23,18)(11,25,26,14)(12,27,29,30)(19,31,33,32)(24,28,34,35)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26350);