Subgroup ($H$) information
| Description: | $C_3\times \He_7$ |
| Order: | \(1029\)\(\medspace = 3 \cdot 7^{3} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Generators: |
$b^{14}, c, d, b^{6}d^{6}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), a semidirect factor, nonabelian, a Hall subgroup, elementary for $p = 7$ (hence hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{21}.D_7^2$ |
| Order: | \(4116\)\(\medspace = 2^{2} \cdot 3 \cdot 7^{3} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_7:(C_6^2:C_2^2)$, of order \(49392\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7^{3} \) |
| $\operatorname{Aut}(H)$ | $(C_7\times C_{14}):\GL(2,7)$, of order \(197568\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $F_7^2:C_2^2$, of order \(7056\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(7\) |
| $W$ | $D_7^2$, of order \(196\)\(\medspace = 2^{2} \cdot 7^{2} \) |
Related subgroups
Other information
| Möbius function | $2$ |
| Projective image | $C_7^2:D_{14}$ |