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×#26401 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
714 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)(14,21)(15,17)(16,20)(18,22)(19,25)(23,26)(24,27);(4,6,8,7)(5,9,10,11)(13,17)(15,25)(18,21)(19,20)(22,23)(26,27);(4,7,8,6)(5,9,10,11)(12,13)(14,18)(15,19)(16,20)(17,25)(21,22)(23,27)(24,26);(12,14,16,24)(13,18,20,26)(15,23,25,22)(17,21,19,27)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 19389);