Subgroup ($H$) information
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$c^{9}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a $p$-group.
Ambient group ($G$) information
| Description: | $C_4^2:C_{12}$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_4:C_{12}$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^3:D_4$, of order \(64\)\(\medspace = 2^{6} \) |
| Outer Automorphisms: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.\GL(2,\mathbb{Z}/4)$, of order \(24576\)\(\medspace = 2^{13} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(S)$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1024\)\(\medspace = 2^{10} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_2\times C_4\times C_{12}$ | |||
| Normalizer: | $C_4^2:C_{12}$ | |||
| Complements: | $C_4:C_{12}$ | |||
| Minimal over-subgroups: | $C_{12}$ | $C_2\times C_4$ | $C_2\times C_4$ | $C_2\times C_4$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $12$ |
| Möbius function | $0$ |
| Projective image | $C_2\times C_4:C_{12}$ |