Subgroup ($H$) information
| Description: | $C_1$ |
| Order: | $1$ |
| Index: | \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \) |
| Exponent: | $1$ |
| Generators: | |
| Nilpotency class: | $0$ |
| Derived length: | $0$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), the Frattini subgroup, cyclic (hence elementary (for every $p$), hyperelementary, metacyclic, and a Z-group), stem, a $p$-group (for every $p$), perfect, and rational. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_2^6.S_3^3$ |
| Order: | \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^6.S_3^3$ |
| Order: | \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^6.S_3^4$, of order \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian, solvable, and rational. 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.S_3^4$, of order \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^6.S_3^3$ |
| Normalizer: | $C_2^6.S_3^3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^6.S_3^3$ |