Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#441 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C19 | C2×D4 |
| commutator | C4 | C22×C4×C19 |
| Frattini, cosocle | C2×C4 | C23×C19 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C42×C19 ◁ G | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,22)(24,25)(26,32)(27,35)(28,30)(29,33)(31,37)(34,39)(36,38);(24,26,28,34)(25,29,30,38)(27,36,37,33)(31,39,35,32);(20,22,23,21)(25,30)(29,38)(31,35)(32,39)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 122);