Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(2519424\)\(\medspace = 2^{7} \cdot 3^{9} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(19,21,23)(20,22,24)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^7.A_4^2:D_{12}$ |
| Order: | \(7558272\)\(\medspace = 2^{7} \cdot 3^{10} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3^7.A_4^2:D_4$ |
| Order: | \(2519424\)\(\medspace = 2^{7} \cdot 3^{9} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Automorphism Group: | $C_3^6.C_2^6:S_3^3$, of order \(10077696\)\(\medspace = 2^{9} \cdot 3^{9} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_2^4.C_3^4.C_2^5.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_3^7.A_4^2:C_{12}$ |
| Normalizer: | $C_3^7.A_4^2:D_{12}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^7.A_4^2:D_{12}$ |