Subgroup ($H$) information
| Description: | $C_{21}:C_6$ |
| Order: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Index: | \(11\) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$a^{3}, b^{154}, b^{66}, a^{2}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_{21}:C_{66}$ |
| Order: | \(1386\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \cdot 11 \) |
| Exponent: | \(462\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{11}$ |
| Order: | \(11\) |
| Exponent: | \(11\) |
| Automorphism Group: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Outer Automorphisms: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| 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, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{10}\times S_3\times F_7$ |
| $\operatorname{Aut}(H)$ | $S_3\times F_7$, of order \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3\times F_7$, of order \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(10\)\(\medspace = 2 \cdot 5 \) |
| $W$ | $C_{21}:C_6$, of order \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_{11}$ | |||
| Normalizer: | $C_{21}:C_{66}$ | |||
| Complements: | $C_{11}$ | |||
| Minimal over-subgroups: | $C_{21}:C_{66}$ | |||
| Maximal under-subgroups: | $C_{21}:C_3$ | $D_{21}$ | $F_7$ | $C_3\times S_3$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{21}:C_{66}$ |