Subgroup ($H$) information
| Description: | $C_2^{13}.A_{14}$ |
| Order: | \(357082280755200\)\(\medspace = 2^{23} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \cdot 11 \cdot 13 \) |
| Index: | $1$ |
| Exponent: | \(720720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \) |
| Generators: |
$\langle(1,16,7,20,11,18,24,5,25,28,9,21,13,2,15,8,19,12,17,23,6,26,27,10,22,14) \!\cdots\! \rangle$
|
| Derived length: | $0$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Hall subgroup, and perfect (hence nonsolvable). Whether it is a direct factor has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{13}.A_{14}$ |
| Order: | \(357082280755200\)\(\medspace = 2^{23} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \cdot 11 \cdot 13 \) |
| Exponent: | \(720720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and perfect (hence nonsolvable).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(714164561510400\)\(\medspace = 2^{24} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \cdot 11 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | Group of order \(714164561510400\)\(\medspace = 2^{24} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \cdot 11 \cdot 13 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |