Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C19 · Schur multiplier (H2) = C2 |
| direct product | C19×#499 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C19 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C19 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C42×C19 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,25)(22,26,27,24)(28,29,31,33)(30,36,37,43)(32,40,39,35)(34,42,41,38);(20,22,23,27)(21,24,25,26)(29,33)(34,41)(35,40)(36,43);(28,30,32,38,31,37,39,42)(29,34,35,36,33,41,40,43)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 180);