Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#12664 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C16 | D4 |
| commutator | C4 | C2×C3×C4×C8 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C22×C3 | C8×D4 |
| prime radical | C23×C3 | #131 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C16 ◁ C22×C3×C16 ◁ G | |
334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(8,9,13,18,14,19,31,28,15,20,32,27,12,17,26,11)(10,21,24,22,23,35,38,37,25,36,30,39,29,16,33,34);(4,5,6,7)(8,10)(9,16)(11,22)(12,29)(13,30)(14,23)(15,25)(17,36)(18,37)(19,21)(20,35)(24,32)(26,38)(27,34)(28,39)(31,33);(8,11,26,17,12,27,32,20,15,28,31,19,14,18,13,9)(10,22,38,36,29,34,24,35,25,39,33,21,23,37,30,16)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5652);