Subgroup ($H$) information
| Description: | $C_3:Q_8$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(215\)\(\medspace = 5 \cdot 43 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$ab^{954}, b^{645}, b^{1290}, b^{1720}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{645}:Q_8$ |
| Order: | \(5160\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 43 \) |
| Exponent: | \(2580\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 43 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $C_{215}$ |
| Order: | \(215\)\(\medspace = 5 \cdot 43 \) |
| Exponent: | \(215\)\(\medspace = 5 \cdot 43 \) |
| Automorphism Group: | $C_2\times C_{84}$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_2\times C_{84}$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 5,43$), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5\times C_5^2:D_{20}$, of order \(8064\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $S_3\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{430}$ | |||
| Normalizer: | $C_{645}:Q_8$ | |||
| Complements: | $C_{215}$ | |||
| Minimal over-subgroups: | $C_{129}:Q_8$ | $C_{15}:Q_8$ | ||
| Maximal under-subgroups: | $C_{12}$ | $C_3:C_4$ | $C_3:C_4$ | $Q_8$ |
Other information
| Möbius function | $1$ |
| Projective image | $D_5^3.C_2^2$ |