Subgroup ($H$) information
| Description: | $C_2\times D_6$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(58\)\(\medspace = 2 \cdot 29 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$a, b^{2}, c^{87}, c^{116}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, an A-group, and rational.
Ambient group ($G$) information
| Description: | $D_6\times C_{116}$ |
| Order: | \(1392\)\(\medspace = 2^{4} \cdot 3 \cdot 29 \) |
| Exponent: | \(348\)\(\medspace = 2^{2} \cdot 3 \cdot 29 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{58}$ |
| Order: | \(58\)\(\medspace = 2 \cdot 29 \) |
| Exponent: | \(58\)\(\medspace = 2 \cdot 29 \) |
| Automorphism Group: | $C_{28}$, of order \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Outer Automorphisms: | $C_{28}$, of order \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,29$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{21}:(C_2^4.C_2^4)$ |
| $\operatorname{Aut}(H)$ | $S_3\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(112\)\(\medspace = 2^{4} \cdot 7 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2\times C_{116}$ | |||||||
| Normalizer: | $D_6\times C_{116}$ | |||||||
| Minimal over-subgroups: | $D_6\times C_{58}$ | $C_4\times D_6$ | ||||||
| Maximal under-subgroups: | $C_2\times C_6$ | $D_6$ | $D_6$ | $D_6$ | $D_6$ | $D_6$ | $D_6$ | $C_2^3$ |
Other information
| Möbius function | $1$ |
| Projective image | $S_3\times C_{58}$ |