Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#504 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C19 | C22 |
| commutator | C2 | C22×C8×C19 |
| Frattini, cosocle | C8 | C23×C19 |
| socle | C2×C19 | C22×C8 |
| prime radical | #41×C19 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C19 ◁ G | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,27,24,28,33,39,25,29,34,40,35,41,45,49)(22,26,31,37,32,38,44,48,30,36,42,46,43,47,50,51);(20,22)(21,26)(23,31)(24,32)(25,30)(27,37)(28,38)(29,36)(33,44)(34,42)(35,43)(39,48)(40,46)(41,47)(45,50)(49,51);(22,30)(26,36)(31,42)(32,43)(37,46)(38,47)(44,50)(48,51)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 185);