Subgroup ($H$) information
| Description: | $C_6.C_6^2.D_4$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Index: | $1$ |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a, e, b^{2}, c^{12}, c^{3}, c^{6}, de^{2}, c^{8}de^{2}, bc^{4}de^{2}$
|
| Derived length: | $4$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, and a Hall subgroup.
Ambient group ($G$) information
| Description: | $C_6.C_6^2.D_4$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_3.C_2^6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $\He_3.C_2^6.C_2^2$ |
| $W$ | $C_6.\SOPlus(4,2)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_2^2$ | |||
| Normalizer: | $C_6.C_6^2.D_4$ | |||
| Complements: | $C_1$ | |||
| Maximal under-subgroups: | $(C_6\times C_{12}).D_6$ | $(C_6\times C_{12}).D_6$ | $(C_2\times \He_3).C_4^2$ | $C_4^2.D_6$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_6.\SOPlus(4,2)$ |