Subgroup ($H$) information
| Description: | $C_7:C_6$ |
| Order: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Index: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$b^{66}, c, b^{88}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
Ambient group ($G$) information
| Description: | $D_{154}:C_6$ |
| Order: | \(1848\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 11 \) |
| Exponent: | \(924\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $D_{22}$ |
| Order: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Automorphism Group: | $C_2\times F_{11}$, of order \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{77}.C_{30}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| $W$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
Related subgroups
| Centralizer: | $D_{22}$ | |||
| Normalizer: | $D_{154}:C_6$ | |||
| Minimal over-subgroups: | $C_7:C_{66}$ | $C_7:C_{12}$ | $C_{14}:C_6$ | $C_2\times F_7$ |
| Maximal under-subgroups: | $C_7:C_3$ | $C_{14}$ | $C_6$ |
Other information
| Möbius function | $-22$ |
| Projective image | $D_{11}\times F_7$ |