Subgroup ($H$) information
| Description: | $C_2\times C_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Index: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
43 & 42 \\
0 & 43
\end{array}\right), \left(\begin{array}{rr}
1 & 42 \\
42 & 1
\end{array}\right), \left(\begin{array}{rr}
49 & 36 \\
48 & 13
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_6^2:(C_2\times S_4)$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_{12}:D_6$ |
| Order: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2\times C_2^4.\SL(3,3)$, of order \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| Outer Automorphisms: | $C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2.C_6^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| $W$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_6^2:(C_2\times S_4)$ |