Finite Group DatabasebetaFinite Group Databasebeta| order | 1232 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7×C11 · Schur multiplier (H2) = C2 |
| direct product | C7×C11×C4⋊C4 |
| unambiguous semidirect product | C4⋊(C4×C7×C11) |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C7×C11 | C22 |
| commutator | C2 | C2×C4×C7×C11 |
| Frattini, cosocle | C22 | C22×C7×C11 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7×C11 ◁ G | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1232 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18)(20,22)(23,24,25,26);(19,20,21,22)770 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1232, 80);