Subgroup ($H$) information
| Description: | $C_6^2:(C_2^2\times A_4)$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(3,6)(5,7)(9,11,10), (10,11)(13,14), (15,16), (9,10,11), (1,2)(3,7)(4,8) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $(S_3^2\times A_4^2):C_2^3$ |
| Order: | \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
| 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: | $C_2\times D_6$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Outer Automorphisms: | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, an A-group, and rational.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{794}:C_{198}$, of order \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $\AGL(2,3)\times C_2^4.(C_6\times A_5).C_2$ |
| $\card{W}$ | \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |