Subgroup ($H$) information
| Description: | $C_2^2\times C_6^2$ |
| Order: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Index: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(20,22)(21,26), (19,25)(23,24), (2,4,12)(3,10,14)(11,15,13), (1,17,9)(2,4,12)(3,10,14)(5,8,16)(6,7,18)(11,15,13), (20,21)(22,26), (19,23)(24,25)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^4.S_4^2:C_2^2$ |
| Order: | \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2\times C_3^4:D_4$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_3^4.Q_8.C_6.C_2^4.C_2$ |
| Outer Automorphisms: | $D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2.A_4^2.D_6^2.C_2^2$ |
| $\operatorname{Aut}(H)$ | $A_8\times \GL(2,3)$, of order \(967680\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |