Subgroup ($H$) information
| Description: | $C_{15}$ |
| Order: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Index: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$d^{2}, b^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{60}.S_4$ |
| Order: | \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $A_4:C_8$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and monomial (hence solvable).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times F_5\times C_3:S_3:S_4$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4320\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $A_4\times C_{60}$ | |||
| Normalizer: | $C_{60}.S_4$ | |||
| Complements: | $A_4:C_8$ $A_4:C_8$ $A_4:C_8$ | |||
| Minimal over-subgroups: | $C_3\times C_{15}$ | $C_{30}$ | $C_{30}$ | $C_{30}$ |
| Maximal under-subgroups: | $C_5$ | $C_3$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{60}.S_4$ |