Subgroup ($H$) information
| Description: | $D_{28}:C_2$ |
| Order: | \(112\)\(\medspace = 2^{4} \cdot 7 \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$a^{3}, c^{7}, c^{14}, b^{3}c^{14}, c^{4}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Hall subgroup, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_{28}.C_6^2$ |
| Order: | \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3^2$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(3\) |
| Automorphism Group: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_{21}\times A_4).C_6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times S_4\times F_7$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times S_4\times F_7$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_6$ | |||
| Normalizer: | $C_{28}.C_6^2$ | |||
| Complements: | $C_3^2$ | |||
| Minimal over-subgroups: | $D_{28}:C_6$ | $D_{28}:C_6$ | ||
| Maximal under-subgroups: | $C_4\times D_7$ | $D_{28}$ | $C_7\times Q_8$ | $D_4:C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $3$ |
| Projective image | $C_{14}:C_6^2$ |