Normal subgroup $C_2\times A_4^2.C_3^2 \trianglelefteq C_6^2:S_4\wr C_2$

• Label: 41472.dq.16.A
• Order: $ 2^{5} \cdot 3^{4} $
• Index: $ 2^{4} $
• Normal: Yes

Subgroup ($H$) information

Description:	not computed
Order:	\(2592\)\(\medspace = 2^{5} \cdot 3^{4} \)
Index:	\(16\)\(\medspace = 2^{4} \)
Exponent:	not computed
Generators:	$\langle(1,3)(2,4)(5,8)(6,7), (1,6)(2,8)(3,7)(4,5), (4,5,8), (4,5,8)(9,13,11)(10,14,12) \!\cdots\! \rangle$
Derived length:	not computed

The subgroup is characteristic (hence normal), nonabelian, metabelian (hence solvable), 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:	$C_6^2:S_4\wr C_2$
Order:	\(41472\)\(\medspace = 2^{9} \cdot 3^{4} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Derived length:	$4$

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

Quotient group ($Q$) structure

Description:	$C_2\times D_4$
Order:	\(16\)\(\medspace = 2^{4} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism Group:	$C_2\wr C_2^2$, of order \(64\)\(\medspace = 2^{6} \)
Outer Automorphisms:	$C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \)
Derived length:	$2$

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

$\operatorname{Aut}(G)$	$C_{1191}:C_{44}$, of order \(995328\)\(\medspace = 2^{12} \cdot 3^{5} \)
$\operatorname{Aut}(H)$	not computed
$W$	$S_4\wr C_2$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)

Related subgroups

Centralizer:	$C_6^2$
Normalizer:	$C_6^2:S_4\wr C_2$
Minimal over-subgroups:	$C_2^2\times A_4^2.C_3^2$	$(C_3\times A_4^2).D_6$	$C_2\times A_4^2.C_3:S_3$	$C_2\times (C_3\times A_4^2).C_6$	$C_6\times A_4^2:S_3$	$(C_3\times A_4^2):C_{12}$
Maximal under-subgroups:	$C_3^2\times A_4^2$	$C_6\times A_4^2$	$C_6\times A_4^2$	$C_2^2:C_6^3$	$C_2^5:C_3^3$	$C_6\times A_4^2$	$C_6\times A_4^2$	$C_6^3:C_3$

Other information

Number of conjugacy classes in this autjugacy class	$1$
Möbius function	not computed
Projective image	$C_3^2:S_4^2:C_2^2$