Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#541 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C19 |
| prime radical | C23×C19 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ G | |
192 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)(22,26)(24,27)(28,29,32,35)(30,33,38,36)(31,39,40,34)(37,42,43,41);(20,21,23,25)(22,24,26,27)(28,29)(30,36)(31,34)(32,35)(33,38)(37,42)(39,40)(41,43);(20,22)(21,24)(23,26)(25,27)(28,30)(29,33)(31,37)(32,38)(34,41)(35,36)(39,42)(40,43);(28,31,32,40)(29,34,35,39)(30,37,38,43)(33,41,36,42)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 222);