Subgroup ($H$) information
| Description: | $C_2\times C_4^2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a^{2}, b, c^{81}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_4^2:C_{108}$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| 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: | $C_{54}$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Automorphism Group: | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Outer Automorphisms: | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.(C_{18}\times A_4).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2^6:S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^6:A_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_2\times C_4\times C_{108}$ | |
| Normalizer: | $C_4^2:C_{108}$ | |
| Minimal over-subgroups: | $C_2\times C_4\times C_{12}$ | $C_4^2:C_4$ |
| Maximal under-subgroups: | $C_2^2\times C_4$ | $C_4^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2^2\times C_{54}$ |