Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#357 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | D8 |
| commutator | C8 | C2×C4×C19 |
| Frattini, cosocle | C2×C8 | C22×C19 |
| prime radical | C2×D8×C19 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C4×C19 ◁ C2×C8×C19 ◁ G | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23)(24,25)(26,29)(27,36)(28,32)(30,33)(31,34)(35,38)(37,39);(25,29)(26,33)(27,34)(30,31)(32,38)(35,39)(36,37)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 38);