Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#560 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C19 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ G | |
252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 304 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,29)(23,28)(24,26)(25,31)(27,33)(30,34)(32,35);(20,22)(21,25)(23,31)(24,32)(26,34)(27,35)(28,29)(30,33);(21,26)(22,30)(25,29)(27,28)(31,32)(34,35);(21,27)(22,31)(25,34)(26,28)(29,35)(30,32)361 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 241);