Subgroup ($H$) information
| Description: | $C_{21}$ |
| Order: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Generators: |
$c^{28}, c^{12}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, and a Hall subgroup.
Ambient group ($G$) information
| Description: | $C_2\times C_4\times C_{84}$ |
| Order: | \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
Quotient group ($Q$) structure
| Description: | $C_2\times C_4^2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^6:S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Outer Automorphisms: | $C_2^6:S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_6\times C_2^6.S_4$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2\times C_4\times C_{84}$ | |
| Normalizer: | $C_2\times C_4\times C_{84}$ | |
| Complements: | $C_2\times C_4^2$ | |
| Minimal over-subgroups: | $C_{42}$ | $C_{42}$ |
| Maximal under-subgroups: | $C_7$ | $C_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2\times C_4^2$ |