Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C25 |
| direct product | C2×C13×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C22 |
| commutator, Frattini, cosocle | C2 | C24×C13 |
| socle | C22×C13 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ G | |
164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(11,12,13,14,15,16,17,18,19,20,21,22,23);(4,7)(8,10);(3,4)(5,8)(6,7)(9,10);(1,2)(3,5,6,9)(4,8,7,10)260 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 230);