Subgroup ($H$) information
| Description: | $(C_2\times C_{12}).D_{18}$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Index: | \(2\) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$ac^{9}, b^{12}, b^{8}, b^{3}c^{9}, c^{18}, c^{12}, c^{28}, b^{6}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_{12}^2.D_6$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_9.(C_2\times C_6^2).C_2^5$ |
| $\operatorname{Aut}(H)$ | $(C_6\times C_{18}).C_6.C_2^5$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $S_3\times D_{18}$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_2\times C_4$ | ||||
| Normalizer: | $C_{12}^2.D_6$ | ||||
| Minimal over-subgroups: | $C_{12}^2.D_6$ | ||||
| Maximal under-subgroups: | $C_{12}:C_{36}$ | $C_{18}:C_{24}$ | $C_{12}.D_{18}$ | $C_{12}.(C_4\times S_3)$ | $C_{18}.C_4^2$ |
Other information
| Möbius function | $-1$ |
| Projective image | $S_3\times D_{18}$ |