Subgroup ($H$) information
| Description: | $C_6.C_2^4$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a, d^{2}, e^{14}, c, d, e^{21}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{84}.C_2^4$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $D_7$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Automorphism Group: | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_3$, of order \(3\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(322560\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2\wr S_5$, of order \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| $\card{W}$ | \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_3\times D_{14}$ | ||
| Normalizer: | $C_{84}.C_2^4$ | ||
| Complements: | $D_7$ $D_7$ | ||
| Minimal over-subgroups: | $C_{84}.C_2^3$ | $C_{12}.C_2^4$ | |
| Maximal under-subgroups: | $C_6\times Q_8$ | $D_4:C_6$ | $C_4.C_2^3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |