Subgroup ($H$) information
| Description: | $C_{20}:C_2^3$ |
| Order: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Index: | \(135\)\(\medspace = 3^{3} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\langle(1,4)(2,3), (1,3)(2,4)(5,9,6,8,7)(11,13)(14,15), (1,3)(2,4), (5,7,8,6,9), (1,4)(2,3)(11,15)(13,14), (1,2)(3,4)(5,8,9,7,6)(10,12)(13,14)\rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_5\times A_4\times A_6$ |
| Order: | \(21600\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times A_6.C_2^2\times S_4$ |
| $\operatorname{Aut}(H)$ | $C_2^7.(C_2^2\times S_4)$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| $W$ | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $45$ |
| Möbius function | $-1$ |
| Projective image | $A_4\times A_6$ |