Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(456\)\(\medspace = 2^{3} \cdot 3 \cdot 19 \) |
| Exponent: | \(2\) |
| Generators: |
$c, d^{19}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $A_4:C_{152}$ |
| Order: | \(1824\)\(\medspace = 2^{5} \cdot 3 \cdot 19 \) |
| Exponent: | \(456\)\(\medspace = 2^{3} \cdot 3 \cdot 19 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_3:C_{152}$ |
| Order: | \(456\)\(\medspace = 2^{3} \cdot 3 \cdot 19 \) |
| Exponent: | \(456\)\(\medspace = 2^{3} \cdot 3 \cdot 19 \) |
| Automorphism Group: | $C_2\times D_6\times C_{18}$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Outer Automorphisms: | $C_2^2\times C_{18}$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_{18}\times S_4$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^2\times C_{76}$ | ||
| Normalizer: | $A_4:C_{152}$ | ||
| Complements: | $C_3:C_{152}$ | ||
| Minimal over-subgroups: | $C_2\times C_{38}$ | $A_4$ | $C_2^3$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $A_4:C_{152}$ |