Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#563 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C19 |
| prime radical | C2×D4×C19 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ G | |
196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,24)(22,26)(28,29,33,31)(30,36,38,34)(32,39,41,35)(37,42,40,43);(21,24)(25,27)(28,30,33,38)(29,34,31,36)(32,40,41,37)(35,43,39,42);(20,21)(22,25)(23,24)(26,27)(28,31,33,29)(30,34,38,36)(32,39,41,35)(37,42,40,43);(20,22)(21,25)(23,26)(24,27)(28,32)(29,35)(30,37)(31,39)(33,41)(34,42)(36,43)(38,40)361 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 244);