Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C4 |
| direct product | C19×#433 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C23 |
| commutator | C22 | C22×C4×C19 |
| Frattini, cosocle | C2×C4 | C23×C19 |
| socle | C22×C19 | C22×C4 |
| prime radical | C23×C19 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,25)(23,24)(26,27)(28,29,31,34,32,35,38,41)(30,33,37,40,36,39,42,43);(21,24)(25,27)(28,30)(29,33)(31,37)(32,36)(34,40)(35,39)(38,42)(41,43);(20,22,23,26)(21,25,24,27)(30,36)(33,39)(37,42)(40,43)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 114);