Subgroup ($H$) information
| Description: | $C_{12}.D_6$ |
| Order: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Index: | \(2\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a, d, c^{4}d^{2}, b^{2}, c^{3}, c^{6}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_4.\SOPlus(4,2)$ |
| Order: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_6^2:C_2^3$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $D_6^2:C_2^3$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_3^2:D_4^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $S_3^2:C_2^2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $S_3^2:C_2^2$ |