Subgroup ($H$) information
| Description: | $C_4:C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a^{3}, c^{3}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_4:C_4\times \He_3$ |
| Order: | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $\He_3$ |
| Order: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Outer Automorphisms: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2:(D_4\times \GL(2,3))$, of order \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_2^2\times \He_3$ | |
| Normalizer: | $C_4:C_4\times \He_3$ | |
| Complements: | $\He_3$ | |
| Minimal over-subgroups: | $C_4:C_{12}$ | $C_4:C_{12}$ |
| Maximal under-subgroups: | $C_2\times C_4$ | $C_2\times C_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2^2\times \He_3$ |