Subgroup ($H$) information
| Description: | $C_5\times A_4$ |
| Order: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Index: | \(17\) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$a, b^{5}c^{17}, c^{17}, b^{2}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, a Hall subgroup, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $A_4\times C_{85}$ |
| Order: | \(1020\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 17 \) |
| Exponent: | \(510\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 17 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{17}$ |
| Order: | \(17\) |
| Exponent: | \(17\) |
| Automorphism Group: | $C_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
| Outer Automorphisms: | $C_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times C_{16}\times S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_4\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_4\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $A_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{85}$ | ||
| Normalizer: | $A_4\times C_{85}$ | ||
| Complements: | $C_{17}$ | ||
| Minimal over-subgroups: | $A_4\times C_{85}$ | ||
| Maximal under-subgroups: | $C_2\times C_{10}$ | $C_{15}$ | $A_4$ |
Other information
| Möbius function | $-1$ |
| Projective image | $A_4\times C_{17}$ |