Subgroup ($H$) information
| Description: | $C_7^2:(C_2^2\times F_7)$ |
| Order: | \(8232\)\(\medspace = 2^{3} \cdot 3 \cdot 7^{3} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$a^{3}, cd^{30}, d^{21}, a^{2}b^{4}d^{14}, b^{21}, d^{6}, b^{6}c^{3}d^{30}$
|
| Derived length: | $3$ |
The subgroup is normal, a semidirect factor, nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $\He_7:C_6^3$ |
| Order: | \(74088\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7^{3} \) |
| 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: | $C_3^2$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(3\) |
| Automorphism Group: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_7.C_3.C_6^3.C_2^3$ |
| $\operatorname{Aut}(H)$ | $\He_7.C_6^2.C_2^3$ |
| $W$ | $\He_7:C_6^2$, of order \(12348\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | $3$ |
| Projective image | $C_{21}.F_7^2$ |