Subgroup ($H$) information
| Description: | $C_1$ |
| Order: | $1$ |
| Index: | \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
| Exponent: | $1$ |
| Generators: | |
| Nilpotency class: | $0$ |
| Derived length: | $0$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), stem (hence central), a $p$-group (for every $p$), perfect, and rational.
Ambient group ($G$) information
| Description: | $C_6^2:S_4\wr C_2$ |
| Order: | \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_6^2:S_4\wr C_2$ |
| Order: | \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $C_{1191}:C_{44}$, of order \(995328\)\(\medspace = 2^{12} \cdot 3^{5} \) |
| Outer Automorphisms: | $S_3\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1191}:C_{44}$, of order \(995328\)\(\medspace = 2^{12} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_6^2:S_4\wr C_2$ | |||||||
| Normalizer: | $C_6^2:S_4\wr C_2$ | |||||||
| Complements: | $C_6^2:S_4\wr C_2$ | |||||||
| Minimal over-subgroups: | $C_2$ | $C_2$ | $C_2$ | $C_2$ | $C_2$ | $C_2$ | $C_2$ | $C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_6^2:S_4\wr C_2$ |