Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×C22⋊Q8 |
| unambiguous semidirect product | C22⋊(Q8×C13) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C13 |
| prime radical | C23×C13 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ G | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,15,16,17,18,19,20,21,22,23,24,25);(5,7,8,12)(6,9,10,11);(2,4)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 183);