Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = B(4,𝔽2) (order 64) · Inn(G) = C22⋊C4 · Out(G) = C22 |
| holomorph | G ⋊ Aut(G) = #72210579 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22⋊C4 |
| commutator | C22 | C2×C4 |
| Frattini, cosocle | C23 | C22 |
| prime radical | C2×D4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 2 generators on 8 points:
(1,2)(3,7,8,5)(4,6);(1,3)(2,5)(4,8)(6,7)Polycyclic presentation:
2^5|a^2=d,b^a=bc,c^a=ce,d^b=de| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 |
| χ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 |
| χ5 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | ||||
| χ6 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | ||||
| χ7 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | ||||
| χ8 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | ||||
| χ9 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ10 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ11 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 6);