Properties

Label 1472.389.8.a1.a1
Order $ 2^{3} \cdot 23 $
Index $ 2^{3} $
Normal Yes

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Subgroup ($H$) information

Description:$C_{184}$
Order: \(184\)\(\medspace = 2^{3} \cdot 23 \)
Index: \(8\)\(\medspace = 2^{3} \)
Exponent: \(184\)\(\medspace = 2^{3} \cdot 23 \)
Generators: $b^{4}, c, b^{16}, b^{8}$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,23$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Ambient group ($G$) information

Description: $C_{23}:Q_{64}$
Order: \(1472\)\(\medspace = 2^{6} \cdot 23 \)
Exponent: \(736\)\(\medspace = 2^{5} \cdot 23 \)
Derived length:$2$

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

Quotient group ($Q$) structure

Description: $D_4$
Order: \(8\)\(\medspace = 2^{3} \)
Exponent: \(4\)\(\medspace = 2^{2} \)
Automorphism Group: $D_4$, of order \(8\)\(\medspace = 2^{3} \)
Outer Automorphisms: $C_2$, of order \(2\)
Nilpotency class: $2$
Derived length: $2$

The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), 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_{184}.C_{44}.C_2^4$
$\operatorname{Aut}(H)$ $C_2^2\times C_{22}$, of order \(88\)\(\medspace = 2^{3} \cdot 11 \)
$\operatorname{res}(\operatorname{Aut}(G))$$C_2^2\times C_{22}$, of order \(88\)\(\medspace = 2^{3} \cdot 11 \)
$\card{\operatorname{ker}(\operatorname{res})}$\(1472\)\(\medspace = 2^{6} \cdot 23 \)
$W$$C_2^2$, of order \(4\)\(\medspace = 2^{2} \)

Related subgroups

Centralizer:$C_{368}$
Normalizer:$C_{23}:Q_{64}$
Minimal over-subgroups:$C_{368}$$Q_{16}\times C_{23}$$C_{23}:Q_{16}$
Maximal under-subgroups:$C_{92}$$C_8$

Other information

Möbius function$0$
Projective image$C_{23}:D_{16}$