Subgroup ($H$) information
Description: | $C_3:S_3$ |
Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
Generators: |
$b^{3}, d^{2}, c^{2}$
|
Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
Ambient group ($G$) information
Description: | $C_6^2:D_6$ |
Order: | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and rational.
Quotient group ($Q$) structure
Description: | $S_4$ |
Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Automorphism Group: | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Outer Automorphisms: | $C_1$, of order $1$ |
Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), 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 $W = G / Z_G(H)$.
$\operatorname{Aut}(G)$ | $C_6^2:D_6$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
$\operatorname{Aut}(H)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
$\operatorname{res}(\operatorname{Aut}(G))$ | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
$\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
$W$ | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
Related subgroups
Centralizer: | $C_2^2$ | ||
Normalizer: | $C_6^2:D_6$ | ||
Complements: | $S_4$ $S_4$ $S_4$ | ||
Minimal over-subgroups: | $C_3^2:C_6$ | $C_6:S_3$ | $S_3^2$ |
Maximal under-subgroups: | $C_3^2$ | $S_3$ | $S_3$ |
Other information
Möbius function | $-12$ |
Projective image | $C_6^2:D_6$ |