Subgroup ($H$) information
| Description: | $C_2^2\times C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$ae^{2}, b, c$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_2^3\times C_4\times C_8$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
Quotient group ($Q$) structure
| Description: | $C_4^2$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $\GL(2,\mathbb{Z}/4)$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Outer Automorphisms: | $\GL(2,\mathbb{Z}/4)$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.C_2^5.C_2^6.C_2^2.\PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8192\)\(\medspace = 2^{13} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^3\times C_4\times C_8$ | |
| Normalizer: | $C_2^3\times C_4\times C_8$ | |
| Minimal over-subgroups: | $C_2^3\times C_4$ | $C_2^3\times C_4$ |
| Maximal under-subgroups: | $C_2\times C_4$ | $C_2^3$ |
Other information
| Number of subgroups in this autjugacy class | $56$ |
| Number of conjugacy classes in this autjugacy class | $56$ |
| Möbius function | $0$ |
| Projective image | $C_4^2$ |