Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#568 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C24 |
| commutator | C2 | C23×C4×C19 |
| Frattini, cosocle | C4 | C24×C19 |
| socle | C2×C19 | C23×C4 |
| prime radical | C2×#41×C19 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ G | |
258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 304 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,27,24,26,30,33)(22,25,29,32,28,31,34,35);(20,22)(21,25)(23,29)(24,28)(26,31)(27,32)(30,34)(33,35);(22,28)(25,31)(29,34)(32,35);(21,26)(25,31)(27,33)(32,35)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 249);