Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#26402 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5,8,10)(6,9,7,11)(12,13,17,21)(14,23,19,26)(15,24,25,18)(16,22,27,20);(4,6,8,7)(5,9,10,11)(12,14,17,19)(13,15,21,25)(16,24,27,18)(20,26,22,23);(4,7,8,6)(5,9,10,11)(12,15)(13,18)(14,22)(16,23)(17,25)(19,20)(21,24)(26,27);(12,15)(13,19)(14,21)(16,26)(17,25)(18,20)(22,24)(23,27)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 19390);