Abstract group $A_4:\SD_{16}$

• Label: 192.973
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{7} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $12$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$A_4:\SD_{16}$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$\GL(2,\mathbb{Z}/4):C_2^2$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \)
Outer automorphisms:	$C_2^2$, of order \(4\)\(\medspace = 2^{2} \)
Composition factors:	$C_2$ x 6, $C_3$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	23	8	56	40	48	16	192
Conjugacy classes	1	5	1	4	3	4	1	19
Divisions	1	5	1	4	2	2	1	16
Autjugacy classes	1	5	1	4	2	2	1	16

Dimension	1	2	3	4	6	12
Irr. complex chars.	4	7	4	1	3	0	19
Irr. rational chars.	4	3	4	3	1	1	16

Minimal Presentations

Permutation degree:	$12$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	$18$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	12
Arbitrary	5	7	7

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{8}=c^{3}=d^{2}=e^{2}=[a,c]=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(2,3)(5,6,8,11)(7,9,12,10), (6,9)(7,12)(10,11), (5,7,8,12)(6,10,11,9), (5,8)(6,11)(7,12)(9,10), (2,3,4), (1,2)(3,4), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 7 & 0 \\ 6 & 7 \end{array}\right), \left(\begin{array}{rr} 0 & 5 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 9 & 10 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 6 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 2 \\ 2 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 2 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/12\Z)$
Transitive group:	24T331	32T2208			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$A_4$ $\,\rtimes\,$ $\SD_{16}$	$(A_4:Q_8)$ $\,\rtimes\,$ $C_2$	$(A_4:C_8)$ $\,\rtimes\,$ $C_2$	$C_2^2$ $\,\rtimes\,$ $(C_3:\SD_{16})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_4$ . $S_4$	$C_4$ . $(C_2\times S_4)$	$(D_4\times A_4)$ . $C_2$	$(C_2\times A_4)$ . $D_4$	all 9

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{12}\Z)$.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 362 subgroups in 80 conjugacy classes, 15 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $\GL(2,\mathbb{Z}/4)$
Commutator:	$G' \simeq$ $C_4\times A_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times D_4$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $A_4:\SD_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_3:D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 192: $A_4:\SD_{16}$

Order 96: $A_4:C_8$, $D_4\times A_4$, $A_4:Q_8$

Order 64: $C_2^2:\SD_{16}$

Order 48: $C_2^2\times A_4$, $C_4\times A_4$, $A_4:C_4$, $C_3:\SD_{16}$

Order 32: $D_4:C_4$ x 2, $C_2\times \SD_{16}$ x 2, $C_2^2:C_8$, $C_2^2\times D_4$, $C_2^2:Q_8$

Order 24: $C_2\times A_4$ x 2, $C_3:Q_8$, $C_3\times D_4$, $C_3:C_8$

Order 16: $\SD_{16}$ x 4, $C_2\times D_4$ x 3, $C_2\times C_8$ x 2, $C_4:C_4$ x 2, $C_2^2:C_4$, $C_2^4$, $C_2\times Q_8$, $C_2^2\times C_4$

Order 12: $C_2\times C_6$, $A_4$, $C_{12}$, $C_3:C_4$

Order 8: $C_2^3$ x 5, $D_4$ x 5, $C_2\times C_4$ x 4, $Q_8$ x 2, $C_8$ x 2

Order 6: $C_6$ x 2

Order 4: $C_2^2$ x 9, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $A_4:\SD_{16}$

Order 96: $A_4:C_8$, $D_4\times A_4$, $A_4:Q_8$

Order 64: $C_2^2:\SD_{16}$

Order 48: $C_2^2\times A_4$, $C_4\times A_4$, $A_4:C_4$, $C_3:\SD_{16}$

Order 32: $D_4:C_4$ x 2, $C_2\times \SD_{16}$ x 2, $C_2^2:C_8$, $C_2^2\times D_4$, $C_2^2:Q_8$

Order 24: $C_2\times A_4$ x 2, $C_3:Q_8$, $C_3\times D_4$, $C_3:C_8$

Order 16: $\SD_{16}$ x 3, $C_2\times D_4$ x 3, $C_2\times C_8$ x 2, $C_4:C_4$ x 2, $C_2^2:C_4$, $C_2^4$, $C_2\times Q_8$, $C_2^2\times C_4$

Order 12: $C_2\times C_6$, $A_4$, $C_{12}$, $C_3:C_4$

Order 8: $C_2^3$ x 5, $D_4$ x 4, $C_2\times C_4$ x 4, $Q_8$ x 2, $C_8$ x 2

Order 6: $C_6$ x 2

Order 4: $C_2^2$ x 9, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 192: $A_4:\SD_{16}$ ($C_1$)

Order 96: $A_4:C_8$ ($C_2$), $D_4\times A_4$ ($C_2$), $A_4:Q_8$ ($C_2$)

Order 48: $C_4\times A_4$ ($C_2^2$)

Order 32: $C_2^2\times D_4$ ($S_3$)

Order 24: $C_2\times A_4$ ($D_4$)

Order 16: $C_2^2\times C_4$ ($D_6$)

Order 12: $A_4$ ($\SD_{16}$)

Order 8: $C_2^3$ ($C_3:D_4$), $D_4$ ($S_4$)

Order 4: $C_2^2$ ($C_3:\SD_{16}$), $C_4$ ($C_2\times S_4$)

Order 2: $C_2$ ($\GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($A_4:\SD_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $A_4:\SD_{16}$ ($C_1$)

Order 96: $A_4:C_8$ ($C_2$), $D_4\times A_4$ ($C_2$), $A_4:Q_8$ ($C_2$)

Order 48: $C_4\times A_4$ ($C_2^2$)

Order 32: $C_2^2\times D_4$ ($S_3$)

Order 24: $C_2\times A_4$ ($D_4$)

Order 16: $C_2^2\times C_4$ ($D_6$)

Order 12: $A_4$ ($\SD_{16}$)

Order 8: $C_2^3$ ($C_3:D_4$), $D_4$ ($S_4$)

Order 4: $C_2^2$ ($C_3:\SD_{16}$), $C_4$ ($C_2\times S_4$)

Order 2: $C_2$ ($\GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($A_4:\SD_{16}$)

Series

Derived series

$A_4:\SD_{16}$

$\rhd$

$C_4\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$A_4:\SD_{16}$

$\rhd$

$D_4\times A_4$

$\rhd$

$C_4\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$A_4:\SD_{16}$

$\rhd$

$C_4\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$D_4$

Supergroups

This group is a maximal subgroup of 28 larger groups in the database.

This group is a maximal quotient of 26 larger groups in the database.

Character theory

Complex character table

See the $19 \times 19$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	2D	2E	3A	4A	4B	4C	4D	6A	6B	8A	8B	12A
	Size	1	1	3	3	4	12	8	2	6	24	24	8	32	24	24	16
	2 P	1A	1A	1A	1A	1A	1A	3A	2A	2A	2A	2C	3A	3A	4B	4A	6A
	3 P	1A	2A	2B	2C	2D	2E	1A	4A	4B	4C	4D	2A	2D	8B	8A	4A
	Schur
192.973.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
192.973.1b		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$
192.973.1c		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$
192.973.1d		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$
192.973.2a		$2$	$2$	$2$	$2$	$2$	$2$	$-1$	$2$	$2$	$0$	$0$	$-1$	$-1$	$0$	$0$	$-1$
192.973.2b		$2$	$2$	$2$	$2$	$0$	$0$	$2$	$-2$	$-2$	$0$	$0$	$2$	$0$	$0$	$0$	$-2$
192.973.2c		$2$	$2$	$2$	$2$	$-2$	$-2$	$-1$	$2$	$2$	$0$	$0$	$-1$	$1$	$0$	$0$	$-1$
192.973.2d		$4$	$-4$	$4$	$-4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$-4$	$0$	$0$	$0$	$0$
192.973.2e		$4$	$4$	$4$	$4$	$0$	$0$	$-2$	$-4$	$-4$	$0$	$0$	$-2$	$0$	$0$	$0$	$2$
192.973.3a		$3$	$3$	$-1$	$-1$	$3$	$-1$	$0$	$3$	$-1$	$1$	$-1$	$0$	$0$	$-1$	$1$	$0$
192.973.3b		$3$	$3$	$-1$	$-1$	$3$	$-1$	$0$	$3$	$-1$	$-1$	$1$	$0$	$0$	$1$	$-1$	$0$
192.973.3c		$3$	$3$	$-1$	$-1$	$-3$	$1$	$0$	$3$	$-1$	$-1$	$1$	$0$	$0$	$-1$	$1$	$0$
192.973.3d		$3$	$3$	$-1$	$-1$	$-3$	$1$	$0$	$3$	$-1$	$1$	$-1$	$0$	$0$	$1$	$-1$	$0$
192.973.4a	2	$4$	$-4$	$4$	$-4$	$0$	$0$	$-2$	$0$	$0$	$0$	$0$	$2$	$0$	$0$	$0$	$0$
192.973.6a		$6$	$6$	$-2$	$-2$	$0$	$0$	$0$	$-6$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
192.973.6b		$12$	$-12$	$-4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$