Subgroup ($H$) information
| Description: | $C_2\times D_6$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,3)(2,4)(5,6)(7,8), (9,11,10), (12,13)(14,15)(16,17)(18,19), (10,11)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, an A-group, and rational. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $S_3\times C_2^6:S_4$ |
| Order: | \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^4:S_4$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^6:(S_3\times \GL(3,2))$, of order \(64512\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 7 \) |
| Outer Automorphisms: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_2^4.S_3^3$ |
| $\operatorname{Aut}(H)$ | $S_3\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\card{W}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^6:S_4$ |
| Normalizer: | $S_3\times C_2^6:S_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |