Subgroup ($H$) information
| Description: | $C_{12}$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Index: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$c^{18}, b^{3}, c^{12}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $\He_3.D_{24}$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3^2:D_6$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_6.S_3^2$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_3.C_3.C_{12}.C_2^5$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{24}.\He_3$ | ||||||
| Normalizer: | $\He_3.D_{24}$ | ||||||
| Minimal over-subgroups: | $C_3\times C_{12}$ | $C_3\times C_{12}$ | $C_{36}$ | $C_{36}$ | $C_{24}$ | $D_{12}$ | $D_{12}$ |
| Maximal under-subgroups: | $C_6$ | $C_4$ |
Other information
| Möbius function | $0$ |
| Projective image | $\He_3.D_{12}$ |