Subgroup ($H$) information
| Description: | $C_7\times Q_8$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$b^{3}c^{14}, c^{4}, c^{14}, c^{7}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_{28}.C_6^2$ |
| Order: | \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3\times C_6$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_{21}\times A_4).C_6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_6\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_6\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| $W$ | $C_2^2\times C_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{42}$ | ||
| Normalizer: | $C_{28}.C_6^2$ | ||
| Complements: | $C_3\times C_6$ | ||
| Minimal over-subgroups: | $Q_8\times C_{21}$ | $C_{28}.C_6$ | $D_{28}:C_2$ |
| Maximal under-subgroups: | $C_{28}$ | $Q_8$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-3$ |
| Projective image | $C_{14}:C_6^2$ |