Abstract group 576.7437: $C_3\times Q_{16}.A_4$

• Label: 576.7437
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $51$
• Trans deg.: $144$
• Rank: $2$

Group information

Description:	$C_3\times Q_{16}.A_4$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$D_8.(D_6\times S_4)$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	31	26	16	86	16	272	128	576
Conjugacy classes	1	4	8	4	14	3	26	18	78
Divisions	1	4	4	4	7	2	13	5	40
Autjugacy classes	1	3	2	3	4	2	5	3	23

Dimension	1	2	3	4	6	8	12	16
Irr. complex chars.	36	9	12	18	3	0	0	0	78
Irr. rational chars.	4	17	4	4	5	1	1	4	40

Minimal presentations

Permutation degree:	$51$
Transitive degree:	$144$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	16
Arbitrary	4	6	10

Constructions

Presentation:	$\langle a, b, c, d \mid c^{2}=d^{24}=[b,d]=[c,d]=1, a^{6}=d^{12}, b^{2}=d^{12}, b^{a}=bcd^{6}, c^{a}=bd^{18}, d^{a}=d^{7}, c^{b}=cd^{12} \rangle$
Permutation group:	Degree $51$ $\langle(1,2,8,13)(3,14,7,12)(4,10,21,30)(5,22,6,11)(9,23,26,16)(15,32,24,31)(17,27,38,45) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $(Q_{16}.A_4)$
Semidirect product:	$(C_8.A_4)$ $\,\rtimes\,$ $C_6$ (3)	$(C_{24}.A_4)$ $\,\rtimes\,$ $C_2$	$(Q_8.C_6^2)$ $\,\rtimes\,$ $C_2$ (2)	$(C_{12}.C_2^4)$ $\,\rtimes\,$ $C_3$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$Q_8$ . $(C_6\times A_4)$ (2)	$C_6$ . $(D_4\times A_4)$	$C_8$ . $(C_6\times A_4)$	$(C_3\times Q_{16})$ . $A_4$	all 22

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{6}^{2} \simeq C_{2}^{2} \times C_{3}^{2}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 690 subgroups in 180 conjugacy classes, 50 normal (22 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $D_4\times A_4$
Commutator:	$G' \simeq$ $D_4:C_2$	$G/G' \simeq$ $C_6^2$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2^2:C_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}.C_2^4$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_3\times Q_{16}.A_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $D_4\times A_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 576: $C_3\times Q_{16}.A_4$

Order 288: $Q_8.C_6^2$ x 2, $C_{24}.A_4$

Order 192: $Q_{16}.A_4$ x 3, $C_{12}.C_2^4$

Order 144: $\SL(2,3):C_6$ x 3, $C_3^2\times Q_{16}$

Order 96: $\SL(2,3):C_2^2$ x 6, $C_8.A_4$ x 3, $C_6.C_2^4$ x 2, $D_8:C_6$ x 2, $D_8:C_6$, $C_6\times D_8$, $\OD_{16}:C_6$

Order 72: $Q_8\times C_3^2$ x 2, $C_3\times \SL(2,3)$, $C_3\times C_{24}$

Order 64: $D_8:C_2^2$

Order 48: $\SL(2,3):C_2$ x 9, $D_4:C_6$ x 5, $C_6\times D_4$ x 4, $C_3\times Q_{16}$ x 4, $C_3\times D_8$ x 3, $C_3\times \SD_{16}$ x 2, $C_3\times \OD_{16}$, $C_2\times C_{24}$

Order 36: $C_3\times C_{12}$ x 3

Order 32: $D_4:C_2^2$ x 2, $D_8:C_2$ x 2, $D_8:C_2$, $C_2\times D_8$, $\OD_{16}:C_2$

Order 24: $C_3\times Q_8$ x 9, $C_3\times D_4$ x 7, $C_{24}$ x 5, $C_2\times C_{12}$ x 3, $\SL(2,3)$ x 3, $C_2^2\times C_6$ x 2

Order 18: $C_3\times C_6$

Order 16: $D_4:C_2$ x 5, $C_2\times D_4$ x 4, $D_8$ x 3, $\SD_{16}$ x 2, $Q_{16}$, $\OD_{16}$, $C_2\times C_8$

Order 12: $C_{12}$ x 13, $C_2\times C_6$ x 5

Order 9: $C_3^2$

Order 8: $D_4$ x 7, $Q_8$ x 3, $C_2\times C_4$ x 3, $C_2^3$ x 2, $C_8$ x 2

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 5, $C_4$ x 4

Order 3: $C_3$ x 4

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $C_3\times Q_{16}.A_4$

Order 288: $Q_8.C_6^2$, $C_{24}.A_4$

Order 192: $C_{12}.C_2^4$, $Q_{16}.A_4$

Order 144: $\SL(2,3):C_6$ x 2, $C_3^2\times Q_{16}$

Order 96: $C_8.A_4$, $C_6.C_2^4$, $\SL(2,3):C_2^2$, $D_8:C_6$, $D_8:C_6$, $C_6\times D_8$, $\OD_{16}:C_6$

Order 72: $Q_8\times C_3^2$, $C_3\times \SL(2,3)$, $C_3\times C_{24}$

Order 64: $D_8:C_2^2$

Order 48: $D_4:C_6$ x 3, $C_6\times D_4$ x 2, $\SL(2,3):C_2$ x 2, $C_3\times Q_{16}$ x 2, $C_3\times D_8$ x 2, $C_3\times \SD_{16}$, $C_3\times \OD_{16}$, $C_2\times C_{24}$

Order 36: $C_3\times C_{12}$ x 2

Order 32: $D_4:C_2^2$, $D_8:C_2$, $D_8:C_2$, $C_2\times D_8$, $\OD_{16}:C_2$

Order 24: $C_3\times D_4$ x 4, $C_{24}$ x 3, $C_3\times Q_8$ x 3, $C_2\times C_{12}$ x 2, $\SL(2,3)$, $C_2^2\times C_6$

Order 18: $C_3\times C_6$

Order 16: $D_4:C_2$ x 3, $D_8$ x 2, $C_2\times D_4$ x 2, $Q_{16}$, $\SD_{16}$, $\OD_{16}$, $C_2\times C_8$

Order 12: $C_{12}$ x 5, $C_2\times C_6$ x 3

Order 9: $C_3^2$

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

Order 6: $C_6$ x 4

Order 4: $C_2^2$ x 3, $C_4$ x 3

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $C_3\times Q_{16}.A_4$ ($C_1$)

Order 288: $Q_8.C_6^2$ ($C_2$) x 2, $C_{24}.A_4$ ($C_2$)

Order 192: $Q_{16}.A_4$ ($C_3$) x 3, $C_{12}.C_2^4$ ($C_3$)

Order 144: $\SL(2,3):C_6$ ($C_2^2$)

Order 96: $\SL(2,3):C_2^2$ ($C_6$) x 6, $C_8.A_4$ ($C_6$) x 3, $C_6.C_2^4$ ($C_6$) x 2, $\OD_{16}:C_6$ ($C_6$)

Order 72: $C_3\times \SL(2,3)$ ($D_4$)

Order 64: $D_8:C_2^2$ ($C_3^2$)

Order 48: $\SL(2,3):C_2$ ($C_2\times C_6$) x 3, $D_4:C_6$ ($C_2\times C_6$), $C_3\times Q_{16}$ ($A_4$)

Order 32: $D_4:C_2^2$ ($C_3\times C_6$) x 2, $\OD_{16}:C_2$ ($C_3\times C_6$)

Order 24: $\SL(2,3)$ ($C_3\times D_4$) x 3, $C_3\times Q_8$ ($C_2\times A_4$) x 2, $C_{24}$ ($C_2\times A_4$), $C_3\times Q_8$ ($C_3\times D_4$)

Order 16: $Q_{16}$ ($C_3\times A_4$), $D_4:C_2$ ($C_6^2$)

Order 12: $C_{12}$ ($C_2^2\times A_4$)

Order 8: $Q_8$ ($C_6\times A_4$) x 2, $Q_8$ ($D_4\times C_3^2$), $C_8$ ($C_6\times A_4$)

Order 6: $C_6$ ($D_4\times A_4$)

Order 4: $C_4$ ($C_2^2:C_6^2$)

Order 3: $C_3$ ($Q_{16}.A_4$)

Order 2: $C_2$ ($C_2^3.C_6^2$)

Order 1: $C_1$ ($C_3\times Q_{16}.A_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $C_3\times Q_{16}.A_4$ ($C_1$)

Order 288: $Q_8.C_6^2$ ($C_2$), $C_{24}.A_4$ ($C_2$)

Order 192: $C_{12}.C_2^4$ ($C_3$), $Q_{16}.A_4$ ($C_3$)

Order 144: $\SL(2,3):C_6$ ($C_2^2$)

Order 96: $C_8.A_4$ ($C_6$), $C_6.C_2^4$ ($C_6$), $\SL(2,3):C_2^2$ ($C_6$), $\OD_{16}:C_6$ ($C_6$)

Order 72: $C_3\times \SL(2,3)$ ($D_4$)

Order 64: $D_8:C_2^2$ ($C_3^2$)

Order 48: $D_4:C_6$ ($C_2\times C_6$), $\SL(2,3):C_2$ ($C_2\times C_6$), $C_3\times Q_{16}$ ($A_4$)

Order 32: $D_4:C_2^2$ ($C_3\times C_6$), $\OD_{16}:C_2$ ($C_3\times C_6$)

Order 24: $\SL(2,3)$ ($C_3\times D_4$), $C_{24}$ ($C_2\times A_4$), $C_3\times Q_8$ ($C_2\times A_4$), $C_3\times Q_8$ ($C_3\times D_4$)

Order 16: $Q_{16}$ ($C_3\times A_4$), $D_4:C_2$ ($C_6^2$)

Order 12: $C_{12}$ ($C_2^2\times A_4$)

Order 8: $Q_8$ ($C_6\times A_4$), $Q_8$ ($D_4\times C_3^2$), $C_8$ ($C_6\times A_4$)

Order 6: $C_6$ ($D_4\times A_4$)

Order 4: $C_4$ ($C_2^2:C_6^2$)

Order 3: $C_3$ ($Q_{16}.A_4$)

Order 2: $C_2$ ($C_2^3.C_6^2$)

Order 1: $C_1$ ($C_3\times Q_{16}.A_4$)

Series

Derived series

$C_3\times Q_{16}.A_4$

$\rhd$

$D_4:C_2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_3\times Q_{16}.A_4$

$\rhd$

$C_{24}.A_4$

$\rhd$

$\OD_{16}:C_6$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3\times Q_{16}.A_4$

$\rhd$

$D_4:C_2$

$\rhd$

$Q_8$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_{12}$

$\lhd$

$C_3\times Q_{16}$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 9 larger groups in the database.

Character theory

Complex character table

See the $78 \times 78$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.