Subgroup ($H$) information
| Description: | $C_3^4.S_4^2:C_4$ |
| Order: | \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \) |
| Index: | $1$ |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a^{3}, c^{3}, e^{3}g^{3}, b^{3}def^{3}g^{5}, d^{2}e, c^{2}d^{4}f^{5}g^{2}, g^{2}, a^{2}c^{4}d^{4}fg^{4}, f^{3}g^{3}, g^{3}, d^{3}f^{3}, b^{2}d^{3}, e^{2}g^{2}, e^{2}f^{2}$
|
| Derived length: | $4$ |
The subgroup is the radical (hence characteristic, normal, and solvable), nonabelian, and a Hall subgroup. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^4.S_4^2:C_4$ |
| Order: | \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^4.D_6\wr C_2$, of order \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_6^4.D_6\wr C_2$, of order \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| $W$ | $C_3^4.S_4^2:C_4$, of order \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \) |
Related subgroups
| Centralizer: | $C_1$ |
| Normalizer: | $C_3^4.S_4^2:C_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4.S_4^2:C_4$ |