Normal subgroup $C_3^3:(S_3\times Q_8) \trianglelefteq C_3^4:(D_4\times \GL(2,3))$

• Label: 31104.mm.24.b1
• Order: $ 2^{4} \cdot 3^{4} $
• Index: $ 2^{3} \cdot 3 $
• Normal: Yes

Subgroup ($H$) information

Description:	$C_3^3:(S_3\times Q_8)$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Index:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Generators:	$\langle(7,12)(8,11)(9,10)(13,15), (1,4)(2,6)(7,13,8,11)(10,12,14,15), (7,13,10) \!\cdots\! \rangle$
Derived length:	$3$

The subgroup is characteristic (hence normal), nonabelian, monomial (hence solvable), and rational.

Ambient group ($G$) information

Description:	$C_3^4:(D_4\times \GL(2,3))$
Order:	\(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Derived length:	$5$

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

Quotient group ($Q$) structure

Description:	$C_2\times D_6$
Order:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism Group:	$S_3\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \)
Outer Automorphisms:	$S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \)
Derived length:	$2$

The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, an A-group, 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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_3:S_3.C_6^2.C_6.D_4.C_2$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \)
$\operatorname{Aut}(H)$	$C_3^4.Q_8^2.S_3^2$, of order \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \)
$W$	$C_3^4:(D_4\times \GL(2,3))$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)

Related subgroups

Centralizer:	$C_1$
Normalizer:	$C_3^4:(D_4\times \GL(2,3))$
Minimal over-subgroups:	$C_3^3:(S_3\times \SL(2,3))$	$S_3^2\times \PSU(3,2)$	$C_3^4:(C_4\times Q_8)$	$(C_3\times F_9):D_6$	$\PSU(3,2):S_3^2$	$C_3^2\wr C_2.\SD_{16}$
Maximal under-subgroups:	$C_3^4:Q_8$	$C_3^3:(C_4\times S_3)$	$C_3^4:Q_8$	$S_3\times \PSU(3,2)$	$C_{12}.D_6$

Other information

Number of conjugacy classes in this autjugacy class	$1$
Möbius function	$24$
Projective image	$C_3^4:(D_4\times \GL(2,3))$