Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×B(4,𝔽2) (order 128) · Inn(G) = C22 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×M16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22 |
| commutator | C2 | C22×C4 |
| Frattini, cosocle | C4 | C23 |
| socle | C22 | C2×C4 |
| prime radical | C23 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 10 points:
(1,2);(3,4,5,8,6,7,9,10);(4,7)(8,10)20 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 37);