Normal subgroup $C_2\times C_2^4.C_3^3:S_3 \trianglelefteq (C_2^3\times C_6^3):D_6$

• Label: 20736.dx.4.D
• Order: $ 2^{6} \cdot 3^{4} $
• Index: $ 2^{2} $
• Normal: Yes

Subgroup ($H$) information

Description:	not computed
Order:	\(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)
Index:	\(4\)\(\medspace = 2^{2} \)
Exponent:	not computed
Generators:	$\langle(12,19,16)(14,15,18), (1,2)(3,5)(4,7)(6,8)(9,11)(12,13,16,17,19,20)(15,18) \!\cdots\! \rangle$
Derived length:	not computed

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

Ambient group ($G$) information

Description:	$(C_2^3\times C_6^3):D_6$
Order:	\(20736\)\(\medspace = 2^{8} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Derived length:	$3$

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

Quotient group ($Q$) structure

Description:	$C_2^2$
Order:	\(4\)\(\medspace = 2^{2} \)
Exponent:	\(2\)
Automorphism Group:	$S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \)
Outer Automorphisms:	$S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \)
Derived length:	$1$

The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_2\times C_7^3:C_6$, of order \(165888\)\(\medspace = 2^{11} \cdot 3^{4} \)
$\operatorname{Aut}(H)$	not computed
$\card{W}$	\(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)

Related subgroups

Centralizer:	$C_2^2$
Normalizer:	$(C_2^3\times C_6^3):D_6$
Minimal over-subgroups:	$C_2^2\times C_2^4.C_3^3:S_3$	$C_2\times C_2^4.(S_3\times \He_3:C_2)$	$C_2^4.(C_6\times \He_3).C_2^2$
Maximal under-subgroups:	$(C_2\times C_6^3).C_6$	$C_2^4.C_3^3:S_3$	$C_2\times C_6^2:S_4$	$C_6^3:D_4$	$C_2\times C_6^2:S_4$	$C_2\times C_6^2:S_4$	$C_6^3:S_3$	$C_6^3:S_3$

Other information

Number of conjugacy classes in this autjugacy class	$1$
Möbius function	not computed
Projective image	not computed