Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#497 |
| 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 | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 608 |
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)(21,23)(24,25,28,31)(26,32,33,39)(27,35,36,30)(29,37,38,34);(24,26,27,34,28,33,36,37)(25,29,30,32,31,38,35,39);(20,21,22,23)(26,33)(29,38)(32,39)(34,37)304 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 178);