Subgroup ($H$) information
| Description: | $C_6\times \GL(3,2)$ |
| Order: | \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(1,3,2)(4,6)(5,7)(8,14,9,12)(10,13), (1,2,3), (4,6)(5,7)(8,10)(11,12), (4,6)(5,7)\rangle$
|
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the socle, nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $D_{12}\times \PSL(2,7)$ |
| Order: | \(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times D_4\times \PGL(2,7)$, of order \(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2\times \PGL(2,7)$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times \PGL(2,7)$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $C_2\times \GL(3,2)$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $2$ |
| Projective image | $D_6\times \GL(3,2)$ |