Subgroup ($H$) information
| Description: | $C_{14}:Q_8$ |
| Order: | \(112\)\(\medspace = 2^{4} \cdot 7 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$a, c^{42}, c^{84}, b^{2}c^{84}, c^{24}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_6.(C_4\times D_{28})$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{42}.(C_2^5\times C_6).C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\wr C_2^2\times F_7$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times D_4\times F_7$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $W$ | $D_{28}$, of order \(56\)\(\medspace = 2^{3} \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $-6$ |
| Projective image | $S_3\times D_{28}$ |