Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#101 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | C22⋊C4 |
| commutator | C22 | C2×C4×C13 |
| Frattini, cosocle | C2×C4 | C22×C13 |
| prime radical | C22×C13 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ G | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18,19,20,21,22,23,24,25,26,27,28,29);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)143 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 51);