Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C2×C19×#125 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C19 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C19 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ G | |
226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37);(5,9)(7,10)(11,12,14,16)(13,15,17,18);(3,4,6,8)(5,7,9,10)(11,12)(13,15)(14,16)(17,18);(3,5)(4,7)(6,9)(8,10)(11,13,14,17)(12,15,16,18)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 208);