Subgroup ($H$) information
| Description: | $C_4^2:A_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,4,2,3)(5,7,6,8)(9,12,10,11)(13,16,14,15)(17,18)(19,20), (1,17,9)(2,18,10) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_4^2.\GL(2,\mathbb{Z}/4)$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $D_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4^2.(D_4\times S_4).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2^6.\POPlus(4,3)$, of order \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \) |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_4^2.\GL(2,\mathbb{Z}/4)$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_4^2.\GL(2,\mathbb{Z}/4)$ |