Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C19 · Schur multiplier (H2) = C4 |
| direct product | C19×#488 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C19 |
| prime radical | C2×D4×C19 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C42×C19 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 608 |
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)(22,25)(23,24)(26,27)(28,29)(30,38)(31,34)(32,42)(33,37)(35,40)(36,39)(41,43);(21,24)(25,27)(28,30,39,41,33,40,32,31)(29,34,42,35,37,43,36,38);(20,22,23,26)(21,25,24,27)(28,31,32,40,33,41,39,30)(29,35,36,34,37,38,42,43)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 169);