Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(2\) |
| Generators: |
$e^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
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, hyperelementary, metacyclic, and a Z-group), stem, a $p$-group, simple, and rational.
Ambient group ($G$) information
| Description: | $C_6^2.S_3^2$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3^4:C_2^3$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $(C_3^2\times C_6^2).\GL(2,3)\wr C_2$, of order \(1492992\)\(\medspace = 2^{11} \cdot 3^{6} \) |
| Outer Automorphisms: | $S_4^2:D_4$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.Q_8^2.D_6^2.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_1$, of order $1$ |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2985984\)\(\medspace = 2^{12} \cdot 3^{6} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_6^2.S_3^2$ | ||||||
| Normalizer: | $C_6^2.S_3^2$ | ||||||
| Minimal over-subgroups: | $C_6$ | $C_6$ | $C_2^2$ | $C_2^2$ | $C_4$ | $C_4$ | $C_4$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-5832$ |
| Projective image | $C_3^4:C_2^3$ |