Subgroup ($H$) information
| Description: | $C_2^9:(C_7^3:\He_3)$ |
| Order: | \(4741632\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 7^{3} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(1,3)(2,8)(4,6)(5,7)(9,10)(11,16)(12,15)(13,14)(17,21)(18,22)(19,23)(20,24) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is the commutator subgroup (hence characteristic and normal), nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^9.C_7^3:C_3\wr S_3$ |
| Order: | \(28449792\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 7^{3} \) |
| Exponent: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^9.C_7^3:C_3\wr S_3$, of order \(28449792\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 7^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2^9.C_7^3:C_3\wr S_3$, of order \(28449792\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 7^{3} \) |
| $W$ | $C_2^9.C_7^3:C_3\wr S_3$, of order \(28449792\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 7^{3} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^9.C_7^3:C_3\wr S_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^9.C_7^3:C_3\wr S_3$ |