Subgroup ($H$) information
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Index: | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$c^{4}, d^{14}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and cyclic (hence elementary ($p = 2,3$), hyperelementary, metacyclic, and a Z-group).
Ambient group ($G$) information
| Description: | $C_{84}.C_2^4$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^3:D_{14}$ |
| Order: | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Automorphism Group: | $F_7\times C_2^6:S_4$, of order \(64512\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 7 \) |
| Outer Automorphisms: | $C_3\times C_2^5:S_4$, of order \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times D_4\times S_4\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\card{W}$ | $1$ |
Related subgroups
| Centralizer: | $C_{84}.C_2^4$ | |||||||
| Normalizer: | $C_{84}.C_2^4$ | |||||||
| Minimal over-subgroups: | $C_{42}$ | $C_2\times C_6$ | $C_{12}$ | $C_{12}$ | $C_2\times C_6$ | $C_2\times C_6$ | $C_{12}$ | $C_{12}$ |
| Maximal under-subgroups: | $C_3$ | $C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |