Subgroup ($H$) information
| Description: | $D_{42}$ |
| Order: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$a, b^{14}, b^{6}, c^{2}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $D_6.D_{42}$ |
| Order: | \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| 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 \(2\) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_6\times C_2^2\times S_3\times F_7$ |
| $\operatorname{Aut}(H)$ | $D_6\times F_7$, of order \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_6\times F_7$, of order \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $D_{42}$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_3:C_4$ | |||
| Normalizer: | $D_6.D_{42}$ | |||
| Minimal over-subgroups: | $C_3\times D_{42}$ | $C_{21}:D_4$ | $C_{21}:D_4$ | $C_4\times D_{21}$ |
| Maximal under-subgroups: | $C_{42}$ | $D_{21}$ | $D_{14}$ | $D_6$ |
Other information
| Möbius function | $-6$ |
| Projective image | $S_3\times D_{42}$ |