Subgroup ($H$) information
| Description: | $C_{21}:(C_3\times S_4)$ |
| Order: | \(1512\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7 \) |
| Index: | \(2\) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$a^{3}b^{3}, c^{3}, b^{2}c^{3}, c^{2}, a^{2}d^{7}, d^{7}, d^{2}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3:S_4\times F_7$ |
| Order: | \(3024\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_7\times C_3:S_3:S_4$ |
| $\operatorname{Aut}(H)$ | $F_7\times C_3:S_3:S_4$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(18144\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_3:S_4\times F_7$, of order \(3024\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_3:S_4\times F_7$ |