Subgroup ($H$) information
| Description: | $C_{12}.\SOPlus(4,2)$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Index: | \(2\) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a, e^{6}, cd, b^{2}e^{9}, b^{4}, e^{4}, de^{4}, e^{3}$
|
| Derived length: | $4$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $(C_4\times \He_3).\SD_{16}$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2:S_3.C_2^4.C_2^3$, of order \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $\He_3.(C_2\times C_4^2).C_2^4$, of order \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \) |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $(C_2\times \He_3).\SD_{16}$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $(C_2\times \He_3).\SD_{16}$ |