Subgroup ($H$) information
| Description: | $C_2^4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(1,3)(2,7)(4,8)(5,6), (2,6)(5,7), (1,4)(2,6)(3,8)(5,7), (2,7)(5,6)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.
Ambient group ($G$) information
| Description: | $C_2^3.\GL(2,\mathbb{Z}/4)$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $A_4:C_4$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and monomial (hence solvable).
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4:D_4\times S_4$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $A_8$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $W$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_2^2.\GL(2,\mathbb{Z}/4)$ |