Subgroup ($H$) information
| Description: | $C_{21}$ |
| Order: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Index: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Exponent: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Generators: |
$c^{70}, c^{15}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{15}:D_{42}$ |
| Order: | \(1260\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $S_3\times C_{10}$ |
| Order: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Automorphism Group: | $C_4\times D_6$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $-1$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times S_3^2\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $S_3\times C_{105}$ | ||||
| Normalizer: | $C_{15}:D_{42}$ | ||||
| Complements: | $S_3\times C_{10}$ | ||||
| Minimal over-subgroups: | $C_{105}$ | $C_3\times C_{21}$ | $D_{21}$ | $C_{42}$ | $D_{21}$ |
| Maximal under-subgroups: | $C_7$ | $C_3$ |
Other information
| Möbius function | $6$ |
| Projective image | $C_{15}:D_{42}$ |