Subgroup ($H$) information
| Description: | $C_2^5\times C_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | $1$ |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a, b, c, e, d, f$
|
| 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), the Fitting subgroup, the radical, a direct factor, a $2$-Sylow subgroup (hence a Hall subgroup), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_2^5\times C_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.C_2^5.\GL(5,2)$, of order \(20478689280\)\(\medspace = 2^{21} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \) |
| $\operatorname{Aut}(H)$ | $C_2^6.C_2^5.\GL(5,2)$, of order \(20478689280\)\(\medspace = 2^{21} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^5\times C_4$ | |
| Normalizer: | $C_2^5\times C_4$ | |
| Complements: | $C_1$ | |
| Maximal under-subgroups: | $C_2^4\times C_4$ | $C_2^6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_1$ |