Subgroup ($H$) information
| Description: | $C_6^2.C_6$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$c^{18}, e^{3}, de^{3}, c^{28}, e^{2}, c^{12}$
|
| Derived length: | $2$ |
The subgroup is the commutator subgroup (hence characteristic and normal), nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $(C_2^2\times D_6):D_{18}$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2.S_4\times C_2^3\times S_3$ |
| $\operatorname{Aut}(H)$ | $C_6^2:S_3^2$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_6\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| $W$ | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $-8$ |
| Projective image | $C_2^2:D_{18}\times S_3$ |