Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C22×C3×C4×S4 (order 1152) · Inn(G) = C22 · Out(G) = C22×C3×C4×D3 |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | C22 |
| commutator | C2 | C22×C4×C13 |
| Frattini, cosocle | C4 | C23×C13 |
| socle | C2×C13 | C22×C4 |
| prime radical | C13×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ G | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16);(3,9)(6,12)(10,15)(13,16)260 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 192);