Subgroup ($H$) information
| Description: | $C_3^8:C_2^3.D_4^2:D_4$ |
| Order: | \(26873856\)\(\medspace = 2^{12} \cdot 3^{8} \) |
| Index: | $1$ |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$\langle(1,7,4)(2,8,5)(3,9,6)(10,15,17)(11,13,18)(12,14,16)(19,24,26)(20,22,27) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a semidirect factor, nonabelian, and a Hall subgroup. Whether it is a direct 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: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^8.C_2.C_4^3.C_2^5.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_3^8.C_2.C_4^3.C_2^5.C_2^2$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| 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 |