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Properties

Label 4608.v.6._.N
Order $ 2^{8} \cdot 3 $
Index $ 2 \cdot 3 $
Normal No

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Subgroup ($H$) information

Description:$C_8.C_{96}$
Order: \(768\)\(\medspace = 2^{8} \cdot 3 \)
Index: \(6\)\(\medspace = 2 \cdot 3 \)
Exponent: \(192\)\(\medspace = 2^{6} \cdot 3 \)
Generators: $\left(\begin{array}{rr} 0 & 5 \\ 5 & 0 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 46 & 0 \\ 0 & 46 \end{array}\right), \left(\begin{array}{rr} 84 & 0 \\ 0 & 144 \end{array}\right), \left(\begin{array}{rr} 81 & 0 \\ 0 & 112 \end{array}\right), \left(\begin{array}{rr} 192 & 0 \\ 0 & 192 \end{array}\right), \left(\begin{array}{rr} 108 & 0 \\ 0 & 108 \end{array}\right), \left(\begin{array}{rr} 186 & 0 \\ 0 & 186 \end{array}\right), \left(\begin{array}{rr} 84 & 0 \\ 0 & 109 \end{array}\right)$ Copy content Toggle raw display
Nilpotency class: $3$
Derived length: $2$

The subgroup is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).

Ambient group ($G$) information

Description: $D_{24}.C_{96}$
Order: \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Exponent: \(192\)\(\medspace = 2^{6} \cdot 3 \)
Derived length:$2$

The ambient group is nonabelian, supersolvable (hence solvable and monomial), 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_4\times C_8).C_2^6.C_2)$
$\operatorname{Aut}(H)$ $(C_2\times C_8).C_2^6$
$\card{\operatorname{res}(S)}$\(1024\)\(\medspace = 2^{10} \)
$\card{\operatorname{ker}(\operatorname{res})}$\(4\)\(\medspace = 2^{2} \)
$W$$D_4$, of order \(8\)\(\medspace = 2^{3} \)

Related subgroups

Centralizer:$C_{192}$
Normalizer:$D_8.C_{96}$
Normal closure:$C_{24}.C_{96}$
Core:$C_4\times C_{96}$
Minimal over-subgroups:$C_{24}.C_{96}$$D_8.C_{96}$
Maximal under-subgroups:$C_4\times C_{96}$$C_3\times \OD_{128}$$C_3\times \OD_{128}$$C_8.C_{32}$
Autjugate subgroups:4608.v.2._.A4608.v.2._.B4608.v.2._.C4608.v.2._.D4608.v.2._.E4608.v.2._.F4608.v.2._.G4608.v.3._.A4608.v.3._.B4608.v.4._.A4608.v.4._.B4608.v.4._.C4608.v.4._.D4608.v.4._.E4608.v.4._.F4608.v.4._.G4608.v.4._.H4608.v.4._.I4608.v.4._.J4608.v.4._.K4608.v.4._.L4608.v.4._.M4608.v.4._.N4608.v.4._.O4608.v.6._.A4608.v.6._.B4608.v.6._.C4608.v.6._.D4608.v.6._.E4608.v.6._.F4608.v.6._.G4608.v.6._.H4608.v.6._.I4608.v.6._.J4608.v.6._.K4608.v.6._.L4608.v.6._.M4608.v.6._.O4608.v.8._.A4608.v.8._.B4608.v.8._.C4608.v.8._.D4608.v.8._.E4608.v.8._.F4608.v.8._.G4608.v.8._.H4608.v.8._.I4608.v.8._.J4608.v.8._.K4608.v.8._.L4608.v.8._.M4608.v.8._.N4608.v.8._.O4608.v.8._.P4608.v.8._.Q4608.v.8._.R4608.v.8._.S4608.v.8._.T4608.v.12._.A4608.v.12._.B4608.v.12._.C4608.v.12._.D4608.v.12._.E4608.v.12._.F4608.v.12._.G4608.v.12._.H4608.v.12._.I4608.v.12._.J4608.v.12._.K4608.v.12._.L4608.v.12._.M4608.v.12._.N4608.v.12._.O4608.v.12._.P4608.v.12._.Q4608.v.12._.R4608.v.12._.S4608.v.12._.T4608.v.12._.U4608.v.12._.V4608.v.12._.W4608.v.12._.X4608.v.12._.Y4608.v.12._.Z4608.v.12._.BA4608.v.12._.BB4608.v.12._.BC4608.v.12._.BD4608.v.12._.BE4608.v.12._.BF4608.v.12._.BG4608.v.16._.A4608.v.16._.B4608.v.16._.C4608.v.16._.D4608.v.16._.E4608.v.16._.F4608.v.16._.G4608.v.16._.H4608.v.16._.I4608.v.16._.J4608.v.16._.K4608.v.16._.L4608.v.16._.M4608.v.16._.N4608.v.16._.O4608.v.16._.P4608.v.16._.Q4608.v.16._.R4608.v.16._.S4608.v.16._.T4608.v.18._.A4608.v.18._.B4608.v.18._.C4608.v.18._.D4608.v.18._.E4608.v.18._.F4608.v.18._.G4608.v.24._.A4608.v.24._.B4608.v.24._.C4608.v.24._.D4608.v.24._.E4608.v.24._.F4608.v.24._.G4608.v.24._.H4608.v.24._.I4608.v.24._.J4608.v.24._.K4608.v.24._.L4608.v.24._.M4608.v.24._.N4608.v.24._.O4608.v.24._.P4608.v.24._.Q4608.v.24._.R4608.v.24._.S4608.v.24._.T4608.v.24._.U4608.v.24._.V4608.v.24._.W4608.v.24._.X4608.v.24._.Y4608.v.24._.Z4608.v.24._.BA4608.v.24._.BB4608.v.24._.BC4608.v.24._.BD4608.v.24._.BE4608.v.24._.BF4608.v.24._.BG4608.v.24._.BH4608.v.24._.BI4608.v.24._.BJ4608.v.24._.BK4608.v.24._.BL4608.v.24._.BM4608.v.24._.BN4608.v.24._.BO4608.v.24._.BP4608.v.24._.BQ4608.v.24._.BR4608.v.24._.BS4608.v.24._.BT4608.v.24._.BU4608.v.32._.A4608.v.32._.B4608.v.32._.C4608.v.32._.D4608.v.32._.E4608.v.32._.F4608.v.32._.G4608.v.32._.H4608.v.32._.I4608.v.32._.J4608.v.32._.K4608.v.32._.L4608.v.32._.M4608.v.32._.N4608.v.32._.O4608.v.32._.P4608.v.32._.Q4608.v.32._.R4608.v.32._.S4608.v.32._.T4608.v.36._.A4608.v.36._.B4608.v.36._.C4608.v.36._.D4608.v.36._.E4608.v.36._.F4608.v.36._.G4608.v.36._.H4608.v.36._.I4608.v.36._.J4608.v.36._.K4608.v.36._.L4608.v.36._.M4608.v.36._.N4608.v.36._.O4608.v.48._.A4608.v.48._.B4608.v.48._.C4608.v.48._.D4608.v.48._.E4608.v.48._.F4608.v.48._.G4608.v.48._.H4608.v.48._.I4608.v.48._.J4608.v.48._.K4608.v.48._.L4608.v.48._.M4608.v.48._.N4608.v.48._.O4608.v.48._.P4608.v.48._.Q4608.v.48._.R4608.v.48._.S4608.v.48._.T4608.v.48._.U4608.v.48._.V4608.v.48._.W4608.v.48._.X4608.v.48._.Y4608.v.48._.Z4608.v.48._.BA4608.v.48._.BB4608.v.48._.BC4608.v.48._.BD4608.v.48._.BE4608.v.48._.BF4608.v.48._.BG4608.v.48._.BH4608.v.48._.BI4608.v.48._.BJ4608.v.48._.BK4608.v.48._.BL4608.v.48._.BM4608.v.48._.BN4608.v.48._.BO4608.v.48._.BP4608.v.48._.BQ4608.v.48._.BR4608.v.48._.BS4608.v.48._.BT4608.v.48._.BU4608.v.64._.A4608.v.64._.B4608.v.64._.C4608.v.64._.D4608.v.64._.E4608.v.64._.F4608.v.64._.G4608.v.64._.H4608.v.64._.I4608.v.64._.J4608.v.64._.K4608.v.64._.L4608.v.64._.M4608.v.64._.N4608.v.64._.O4608.v.64._.P4608.v.72._.A4608.v.72._.B4608.v.72._.C4608.v.72._.D4608.v.72._.E4608.v.72._.F4608.v.72._.G4608.v.72._.H4608.v.72._.I4608.v.72._.J4608.v.72._.K4608.v.72._.L4608.v.72._.M4608.v.72._.N4608.v.72._.O4608.v.72._.P4608.v.72._.Q4608.v.72._.R4608.v.72._.S4608.v.72._.T4608.v.96._.A4608.v.96._.B4608.v.96._.C4608.v.96._.D4608.v.96._.E4608.v.96._.F4608.v.96._.G4608.v.96._.H4608.v.96._.I4608.v.96._.J4608.v.96._.K4608.v.96._.L4608.v.96._.M4608.v.96._.N4608.v.96._.O4608.v.96._.P4608.v.96._.Q4608.v.96._.R4608.v.96._.S4608.v.96._.T4608.v.96._.U4608.v.96._.V4608.v.96._.W4608.v.96._.X4608.v.96._.Y4608.v.96._.Z4608.v.96._.BA4608.v.96._.BB4608.v.96._.BC4608.v.96._.BD4608.v.96._.BE4608.v.96._.BF4608.v.96._.BG4608.v.96._.BH4608.v.96._.BI4608.v.96._.BJ4608.v.96._.BK4608.v.96._.BL4608.v.96._.BM4608.v.96._.BN4608.v.96._.BO4608.v.96._.BP4608.v.96._.BQ4608.v.96._.BR4608.v.96._.BS4608.v.96._.BT4608.v.96._.BU4608.v.128._.A4608.v.128._.B4608.v.128._.C4608.v.128._.D4608.v.128._.E4608.v.128._.F4608.v.128._.G4608.v.128._.H4608.v.128._.I4608.v.144._.A4608.v.144._.B4608.v.144._.C4608.v.144._.D4608.v.144._.E4608.v.144._.F4608.v.144._.G4608.v.144._.H4608.v.144._.I4608.v.144._.J4608.v.144._.K4608.v.144._.L4608.v.144._.M4608.v.144._.N4608.v.144._.O4608.v.144._.P4608.v.144._.Q4608.v.144._.R4608.v.144._.S4608.v.144._.T4608.v.192._.A4608.v.192._.B4608.v.192._.C4608.v.192._.D4608.v.192._.E4608.v.192._.F4608.v.192._.G4608.v.192._.H4608.v.192._.I4608.v.192._.J4608.v.192._.K4608.v.192._.L4608.v.192._.M4608.v.192._.N4608.v.192._.O4608.v.192._.P4608.v.192._.Q4608.v.192._.R4608.v.192._.S4608.v.192._.T4608.v.192._.U4608.v.192._.V4608.v.192._.W4608.v.192._.X4608.v.192._.Y4608.v.192._.Z4608.v.192._.BA4608.v.192._.BB4608.v.192._.BC4608.v.192._.BD4608.v.192._.BE4608.v.192._.BF4608.v.192._.BG4608.v.192._.BH4608.v.192._.BI4608.v.192._.BJ4608.v.192._.BK4608.v.192._.BL4608.v.192._.BM4608.v.256._.A4608.v.256._.B4608.v.256._.C4608.v.256._.D4608.v.288._.A4608.v.288._.B4608.v.288._.C4608.v.288._.D4608.v.288._.E4608.v.288._.F4608.v.288._.G4608.v.288._.H4608.v.288._.I4608.v.288._.J4608.v.288._.K4608.v.288._.L4608.v.288._.M4608.v.288._.N4608.v.288._.O4608.v.288._.P4608.v.288._.Q4608.v.288._.R4608.v.288._.S4608.v.288._.T4608.v.384._.A4608.v.384._.B4608.v.384._.C4608.v.384._.D4608.v.384._.E4608.v.384._.F4608.v.384._.G4608.v.384._.H4608.v.384._.I4608.v.384._.J4608.v.384._.K4608.v.384._.L4608.v.384._.M4608.v.384._.N4608.v.384._.O4608.v.384._.P4608.v.384._.Q4608.v.384._.R4608.v.384._.S4608.v.384._.T4608.v.384._.U4608.v.384._.V4608.v.384._.W4608.v.384._.X4608.v.384._.Y4608.v.576._.A4608.v.576._.B4608.v.576._.C4608.v.576._.D4608.v.576._.E4608.v.576._.F4608.v.576._.G4608.v.576._.H4608.v.576._.I4608.v.576._.J4608.v.576._.K4608.v.576._.L4608.v.576._.M4608.v.576._.N4608.v.576._.O4608.v.576._.P4608.v.768._.A4608.v.768._.B4608.v.768._.C4608.v.768._.D4608.v.768._.E4608.v.768._.F4608.v.768._.G4608.v.768._.H4608.v.768._.I4608.v.768._.J4608.v.768._.K4608.v.1152._.A4608.v.1152._.B4608.v.1152._.C4608.v.1152._.D4608.v.1152._.E4608.v.1152._.F4608.v.1152._.G4608.v.1152._.H4608.v.1152._.I4608.v.1536._.A4608.v.1536._.B4608.v.1536._.C4608.v.2304._.A4608.v.2304._.B4608.v.2304._.C4608.v.2304._.D

Other information

Number of subgroups in this conjugacy class$3$
Möbius function$1$
Projective image$C_2\times D_{12}$