Properties

Label 324000.bm.3240.h1
Order $ 2^{2} \cdot 5^{2} $
Index $ 2^{3} \cdot 3^{4} \cdot 5 $
Normal No

Downloads

Learn more

Subgroup ($H$) information

Description:$D_5^2$
Order: \(100\)\(\medspace = 2^{2} \cdot 5^{2} \)
Index: \(3240\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 5 \)
Exponent: \(10\)\(\medspace = 2 \cdot 5 \)
Generators: $acd^{15}e^{11}f^{10}, d^{6}, e^{3}f^{3}, b^{6}d^{15}e^{5}f^{4}$ Copy content Toggle raw display
Derived length: $2$

The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Ambient group ($G$) information

Description: $C_{15}^3.(C_4\times S_4)$
Order: \(324000\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5^{3} \)
Exponent: \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Derived length:$4$

The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.

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)$$D_{15}\wr S_3.C_4$, of order \(648000\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{3} \)
$\operatorname{Aut}(H)$ $F_5\wr C_2$, of order \(800\)\(\medspace = 2^{5} \cdot 5^{2} \)
$W$$D_5:F_5$, of order \(200\)\(\medspace = 2^{3} \cdot 5^{2} \)

Related subgroups

Centralizer:$C_3\times D_5$
Normalizer:$C_5^3.C_6.C_2^3$
Normal closure:$C_3^3:D_5\wr S_3$
Core:$C_1$
Minimal over-subgroups:$C_5\times D_5^2$$C_3\times D_5^2$$D_5\times D_{15}$$C_{15}:D_{10}$$D_5\times D_{10}$
Maximal under-subgroups:$C_5\times D_5$$C_5\times D_5$$C_5:D_5$$D_{10}$$D_{10}$

Other information

Number of subgroups in this autjugacy class$54$
Number of conjugacy classes in this autjugacy class$1$
Möbius function not computed
Projective image$C_{15}^3.(C_4\times S_4)$