Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#327 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C22⋊C4 |
| commutator | C2×C4 | C2×C4×C19 |
| Frattini, cosocle | C22×C4 | C22×C19 |
| prime radical | C19×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 304 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,25,26,33)(22,29)(27,34,28,30)(31,35);(20,21)(22,30)(23,27)(24,28)(25,31)(26,32)(29,33)(34,35)361 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 8);