Subgroup ($H$) information
| Description: | $C_{12}$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Index: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a^{2}b^{66}, b^{176}, b^{132}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $C_{264}:C_4$ |
| Order: | \(1056\)\(\medspace = 2^{5} \cdot 3 \cdot 11 \) |
| Exponent: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $C_4\times D_{11}$ |
| Order: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Exponent: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Automorphism Group: | $C_2^2\times F_{11}$, of order \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^3\times C_{11}:C_5).C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3520\)\(\medspace = 2^{6} \cdot 5 \cdot 11 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{264}:C_4$ | |||
| Normalizer: | $C_{264}:C_4$ | |||
| Minimal over-subgroups: | $C_{132}$ | $C_2\times C_{12}$ | $C_{24}$ | $C_{24}$ |
| Maximal under-subgroups: | $C_6$ | $C_4$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_4\times D_{11}$ |