Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#20787 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | #117 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | #2283 |
| prime radical | C2×C3×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C3×#405 ◁ G | |
616 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)(6,15,18,11)(7,12)(8,13)(9,14)(10,17,23,31)(16,28,30,24)(19,25)(20,26)(21,27)(22,29,33,35)(32,34);(4,6)(5,10,14,23)(7,16)(8,17)(9,18)(11,13,15,27)(12,22,26,33)(19,29)(20,30)(21,31)(24,25,28,34)(32,35);(4,6)(5,11)(7,16)(8,17)(9,18)(10,13)(12,24)(14,15)(19,29)(20,30)(21,31)(22,25)(23,27)(26,28)(32,35)(33,34);(4,7,8,19,9,20,21,32)(5,12,13,25,14,26,27,34)(6,16,17,29,18,30,31,35)(10,22,15,28,23,33,11,24)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13775);