Subgroup ($H$) information
| Description: | $C_3^8:(C_2^2\times C_4)$ |
| Order: | \(104976\)\(\medspace = 2^{4} \cdot 3^{8} \) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(19,20,21)(22,23,24)(25,26,27)(28,31,34)(29,32,35)(30,33,36), (1,3)(4,9) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, metabelian (hence solvable), and an A-group. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8:C_4^2.Q_{16}:C_2^2$ |
| Order: | \(6718464\)\(\medspace = 2^{10} \cdot 3^{8} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^3:D_4$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^9.S_4$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| Outer Automorphisms: | $C_2^6:S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^8:C_4^2.D_4^2:D_4$, of order \(53747712\)\(\medspace = 2^{13} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $C_3^8:C_8.D_8^2.\SD_{16}.S_4$, of order \(5159780352\)\(\medspace = 2^{18} \cdot 3^{9} \) |
| $\card{W}$ | 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 |