Subgroup ($H$) information
| Description: | $C_{12}.S_3^2$ |
| Order: | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$ac^{3}e^{6}, d^{2}, c^{2}, b, e^{4}, e^{3}, e^{6}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_{12}.D_6^2$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3\wr C_4:D_6$, of order \(55296\)\(\medspace = 2^{11} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $C_3^3.C_2^6.C_2^2$ |
| $\card{W}$ | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_4$ | |||||
| Normalizer: | $C_{12}.D_6^2$ | |||||
| Minimal over-subgroups: | $C_6.D_6^2$ | $C_4.S_3^3$ | ||||
| Maximal under-subgroups: | $C_3:S_3\times C_{12}$ | $C_6.S_3^2$ | $C_3^3:Q_8$ | $C_3^3:Q_8$ | $C_{12}.D_6$ | $C_{12}.D_6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |