Subgroup ($H$) information
| Description: | $C_4^2:C_{108}$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Index: | $1$ |
| Exponent: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Generators: |
$a, c^{36}, a^{2}, b^{2}c^{54}, b, c^{76}, c^{12}, c^{54}, c^{27}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the radical, a direct factor, nonabelian, a Hall subgroup, elementary for $p = 2$ (hence hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_4^2:C_{108}$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $0$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.(C_{18}\times A_4).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2^8.(C_{18}\times A_4).C_2$ |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_2^2\times C_{54}$ | ||
| Normalizer: | $C_4^2:C_{108}$ | ||
| Complements: | $C_1$ | ||
| Maximal under-subgroups: | $C_{54}.C_4^2$ | $C_2\times C_4\times C_{108}$ | $C_4^2:C_{36}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_2^3$ |