Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Exponent: | \(2\) |
| Generators: |
$ab, c^{2}d^{30}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $C_2\times C_{60}.D_4$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_{30}:Q_8$ |
| Order: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Automorphism Group: | $(D_6\times C_4^2):D_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Outer Automorphisms: | $C_2^3.C_2^4$, of order \(128\)\(\medspace = 2^{7} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^9.C_2^6)$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{W}$ | \(2\) |
Related subgroups
| Centralizer: | $C_2^3\times C_{60}$ | |||
| Normalizer: | $C_2\times C_{60}.D_4$ | |||
| Complements: | $C_{30}:Q_8$ | |||
| Minimal over-subgroups: | $C_2\times C_{10}$ | $C_2\times C_6$ | $C_2^3$ | $C_2^3$ |
| Maximal under-subgroups: | $C_2$ | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | not computed |