Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#407 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C23 |
| commutator | C22 | C22×C4×C19 |
| Frattini, cosocle | C2×C4 | C23×C19 |
| socle | C22×C19 | C22×C4 |
| prime radical | C24×C19 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ G | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 304 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,23)(24,25,26,29,27,28,30,31);(21,23);(25,28)(29,31)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 88);