Properties

Label 296448.l.3088.h1.b1
Order $ 2^{5} \cdot 3 $
Index $ 2^{4} \cdot 193 $
Normal No

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Subgroup ($H$) information

Description:$C_{96}$
Order: \(96\)\(\medspace = 2^{5} \cdot 3 \)
Index: \(3088\)\(\medspace = 2^{4} \cdot 193 \)
Exponent: \(96\)\(\medspace = 2^{5} \cdot 3 \)
Generators: $a^{12}b^{12009}, b^{9264}, b^{4632}, b^{6176}, a^{8}b^{13930}, b^{2316}$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Ambient group ($G$) information

Description: $C_{18528}.C_{16}$
Order: \(296448\)\(\medspace = 2^{9} \cdot 3 \cdot 193 \)
Exponent: \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Derived length:$2$

The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.

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_{1544}.C_{24}.C_4^2.C_2^5$
$\operatorname{Aut}(H)$ $C_2^2\times C_8$, of order \(32\)\(\medspace = 2^{5} \)
$W$$C_1$, of order $1$

Related subgroups

Centralizer:$C_8\times C_{192}$
Normalizer:$C_8\times C_{192}$
Normal closure:$C_3\times C_{193}:C_{32}$
Core:$C_{24}$
Minimal over-subgroups:$C_3\times C_{193}:C_{32}$$C_2\times C_{96}$
Maximal under-subgroups:$C_{48}$$C_{32}$
Autjugate subgroups:296448.l.3088.h1.a1

Other information

Number of subgroups in this conjugacy class$193$
Möbius function$0$
Projective image$C_{772}:C_{16}$