Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C23×C13 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ G | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 104 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(3,5)(4,6)(7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21);(2,5)(3,8)(6,7);(2,6)(5,7)143 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 197);