Subgroup ($H$) information
| Description: | $C_5$ |
| Order: | \(5\) |
| Index: | \(372\)\(\medspace = 2^{2} \cdot 3 \cdot 31 \) |
| Exponent: | \(5\) |
| Generators: |
$b^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a direct factor, cyclic (hence elementary, hyperelementary, metacyclic, and a Z-group), a $5$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{155}:A_4$ |
| Order: | \(1860\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 31 \) |
| Exponent: | \(930\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{31}:A_4$ |
| Order: | \(372\)\(\medspace = 2^{2} \cdot 3 \cdot 31 \) |
| Exponent: | \(186\)\(\medspace = 2 \cdot 3 \cdot 31 \) |
| Automorphism Group: | $C_{31}:(C_{30}\times A_4)$ |
| Outer Automorphisms: | $C_{30}$, of order \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{31}.(C_{30}\times A_4).C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(11160\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 31 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{155}:A_4$ | ||
| Normalizer: | $C_{155}:A_4$ | ||
| Complements: | $C_{31}:A_4$ | ||
| Minimal over-subgroups: | $C_{155}$ | $C_{15}$ | $C_{10}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $-124$ |
| Projective image | $C_{31}:A_4$ |