Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#341 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | D4 |
| commutator | C4 | C42×C19 |
| Frattini, cosocle | C22×C4 | C22×C19 |
| socle | C22×C19 | C4⋊C4 |
| prime radical | C23×C19 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×C4×C19 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,23)(24,25,27,32,29,33,28,31)(26,34,35,30,37,38,36,39);(21,23)(24,26,28,36,29,37,27,35)(25,30,32,38,33,39,31,34)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 22);