Subgroup ($H$) information
| Description: | $C_2^4:A_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(4,12,5)(6,11,9)(7,8,10), (1,12)(3,9)(4,5)(6,11), (1,5)(4,12), (1,4)(2,10)(3,11)(5,12)(6,9)(7,8), (3,9)(6,11), (3,6)(9,11), (1,5)(2,7)(4,12)(8,10)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, monomial (hence solvable), metabelian, and an A-group. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $A_4^3.(C_4\times S_4)$ |
| Order: | \(165888\)\(\medspace = 2^{11} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3^2:(C_4\times S_4)$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^2\times C_6^2:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $3$ |
The quotient is nonabelian and monomial (hence solvable).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^3.(C_2^3\times S_4)$, of order \(331776\)\(\medspace = 2^{12} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $\AGammaL(3,4)$, of order \(23224320\)\(\medspace = 2^{13} \cdot 3^{4} \cdot 5 \cdot 7 \) |
| $W$ | $A_4^3:D_6$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_2^3$ |
| Normalizer: | $A_4^3.(C_4\times S_4)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $A_4^3.(C_4\times S_4)$ |