Normal subgroup $C_3^2\times C_5^2:D_5 \trianglelefteq D_5^3.C_3^2:D_6$

• Label: 108000.b.48.b1
• Order: $ 2 \cdot 3^{2} \cdot 5^{3} $
• Index: $ 2^{4} \cdot 3 $
• Normal: Yes

Subgroup ($H$) information

Description:	not computed
Order:	\(2250\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{3} \)
Index:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Exponent:	not computed
Generators:	$a^{2}c^{4}d^{25}ef^{3}, f, c^{2}d^{20}, d^{6}, d^{20}, ef$
Derived length:	not computed

The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.

Ambient group ($G$) information

Description:	$D_5^3.C_3^2:D_6$
Order:	\(108000\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{3} \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Derived length:	$4$

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

Quotient group ($Q$) structure

Description:	$C_2\times S_4$
Order:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism Group:	$C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \)
Outer Automorphisms:	$C_2$, of order \(2\)
Derived length:	$3$

The quotient is nonabelian, monomial (hence solvable), 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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$D_5^3:\He_3.C_2^3$, of order \(216000\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{3} \)
$\operatorname{Aut}(H)$	not computed
$W$	$D_5^3.D_6$, of order \(12000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{3} \)

Related subgroups

Centralizer:	$C_3^2$
Normalizer:	$D_5^3.C_3^2:D_6$
Minimal over-subgroups:	$(C_5\times C_{15}^2):C_6$	$C_5^3:(C_2\times C_3:S_3)$	$C_3^2\times D_5\times C_5:D_5$	$C_5^3:(C_2\times C_3:S_3)$	$C_3\times C_5^3:(C_3:C_4)$	$C_3\times C_5^3:(C_3:C_4)$
Maximal under-subgroups:	$C_5\times C_{15}^2$	$C_5^3:C_6$	$C_5^3:C_6$	$C_{15}^2:C_2$	$C_{15}^2:C_2$	$C_{15}^2:C_2$	$C_{15}^2:C_2$	$C_{15}^2:C_2$

Other information

Number of conjugacy classes in this autjugacy class	$1$
Möbius function	$24$
Projective image	$D_5^3.C_3^2:D_6$