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: |
$a^{3}b^{3}c^{3}, a^{6}c^{9}, c^{6}, c^{3}, c^{4}, b^{6}, b^{4}, a^{4}b^{6}c^{6}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_4.D_8\times \He_3$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The ambient group is nonabelian, nilpotent (hence solvable, supersolvable, 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 $1$ |
| Nilpotency class: | $1$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times D_4^2\times \AGL(2,3)$ |
| $\operatorname{Aut}(H)$ | $\ASL(2,3).C_2^5.C_2^2$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $D_4\times C_3^2$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_2\times C_{12}$ | ||
| Normalizer: | $C_4.D_8\times \He_3$ | ||
| Minimal over-subgroups: | $C_4.D_8\times \He_3$ | ||
| Maximal under-subgroups: | $C_4^2\times \He_3$ | $C_{12}.C_6^2$ | $C_{12}:C_{24}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_6^2:C_4$ |