Subgroup ($H$) information
| Description: | $C_3^8:C_4^2$ |
| Order: | \(104976\)\(\medspace = 2^{4} \cdot 3^{8} \) |
| Index: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,9,5)(2,7,6)(3,8,4)(19,24,26)(20,22,27)(21,23,25), (28,30,29)(31,33,32) \!\cdots\! \rangle$
|
| 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: | $C_3^8:C_2^3.D_4^2:D_4$ |
| Order: | \(26873856\)\(\medspace = 2^{12} \cdot 3^{8} \) |
| 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: | $(C_2^2\times C_8):C_2^3$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2\times D_4^3.D_6$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| Outer Automorphisms: | $C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^8.C_2.C_4^3.C_2^5.C_2$ |
| $\operatorname{Aut}(H)$ | $C_3^8:C_8.D_8^2.\SD_{16}.C_2^2$, of order \(859963392\)\(\medspace = 2^{17} \cdot 3^{8} \) |
| $\card{W}$ | \(26873856\)\(\medspace = 2^{12} \cdot 3^{8} \) |
Related subgroups
| Centralizer: | $C_1$ |
| Normalizer: | $C_3^8:C_2^3.D_4^2:D_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |