Subgroup ($H$) information
| Description: | $C_{21}$ |
| Order: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Generators: |
$e^{14}, e^{6}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a Hall subgroup.
Ambient group ($G$) information
| Description: | $C_{84}.C_2^4$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_4.C_2^4$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^8.(C_2\times S_5)$, of order \(61440\)\(\medspace = 2^{12} \cdot 3 \cdot 5 \) |
| Outer Automorphisms: | $C_2\wr S_5$, of order \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(322560\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{W}$ | \(2\) |
Related subgroups
| Centralizer: | $C_{84}.C_2^3$ | |||
| Normalizer: | $C_{84}.C_2^4$ | |||
| Complements: | $C_4.C_2^4$ | |||
| Minimal over-subgroups: | $C_{42}$ | $C_3\times D_7$ | $C_{42}$ | $C_3\times D_7$ |
| Maximal under-subgroups: | $C_7$ | $C_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |