Subgroup ($H$) information
| Description: | $C_2\times C_{28}$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$c, b^{4}d^{6}, d^{6}, d^{9}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
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: | $C_3:D_4$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2\times D_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{42}.(C_2^4\times C_6).C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_6\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\card{W}$ | \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |