Subgroup ($H$) information
| Description: | $C_{12}^2.C_6$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Index: | \(2\) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a^{3}, c^{6}, b^{6}, c^{9}, a^{6}, c^{4}, b^{4}, a^{4}b^{6}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_{12}^2.D_6$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\GL(2,3).C_2^6$, of order \(27648\)\(\medspace = 2^{10} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $\He_3.C_2^6.C_2^2$ |
| $\card{W}$ | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_2\times C_4$ | |||
| Normalizer: | $C_{12}^2.D_6$ | |||
| Minimal over-subgroups: | $C_{12}^2.D_6$ | |||
| Maximal under-subgroups: | $C_4^2\times \He_3$ | $C_6^2.C_{12}$ | $C_{12}.D_{12}$ | $C_{12}:C_{24}$ |
Other information
| Möbius function | not computed |
| Projective image | not computed |