Subgroup ($H$) information
| Description: | $C_7^2$ |
| Order: | \(49\)\(\medspace = 7^{2} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(7\) |
| Generators: |
$\left(\begin{array}{rrr}
1 & 4 & 0 \\
0 & 1 & 0 \\
0 & 2 & 1
\end{array}\right), \left(\begin{array}{rrr}
5 & 0 & 6 \\
0 & 1 & 0 \\
2 & 0 & 4
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the socle, a semidirect factor, abelian (hence metabelian and an A-group), a $7$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_7^2:S_3$ |
| Order: | \(294\)\(\medspace = 2 \cdot 3 \cdot 7^{2} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and an A-group.
Quotient group ($Q$) structure
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^2:(C_6\times S_3)$, of order \(1764\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $\GL(2,7)$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_6\times S_3$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(49\)\(\medspace = 7^{2} \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_7^2$ | ||
| Normalizer: | $C_7^2:S_3$ | ||
| Complements: | $S_3$ | ||
| Minimal over-subgroups: | $C_7^2:C_3$ | $C_7\times D_7$ | |
| Maximal under-subgroups: | $C_7$ | $C_7$ | $C_7$ |
Other information
| Möbius function | $3$ |
| Projective image | $C_7^2:S_3$ |