Subgroup ($H$) information
| Description: | $C_2^2\times C_{12}$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
15 & 14 \\
14 & 15
\end{array}\right), \left(\begin{array}{rr}
1 & 14 \\
14 & 1
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 13
\end{array}\right), \left(\begin{array}{rr}
20 & 21 \\
21 & 6
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 9
\end{array}\right)$
|
| 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), and elementary for $p = 2$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $C_6^2:C_2^4$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| 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: | $D_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, an A-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^7.C_2^4.D_6^2$ |
| $\operatorname{Aut}(H)$ | $C_2^4:S_4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^4:S_4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-6$ |
| Projective image | $C_2\times D_6$ |