Abstract group 1728.18214: $C_9\times Q_{16}.A_4$

• Label: 1728.18214
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{2} \)
• Perm deg.: $57$
• Trans deg.: $432$
• Rank: $2$

Group information

Description:	$C_9\times Q_{16}.A_4$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$(C_{12}\times A_4).C_6.C_2^4$, of order \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Derived length:	$3$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	6	8	9	12	18	24	36	72
Elements	1	31	26	16	86	16	54	272	234	128	576	288	1728
Conjugacy classes	1	4	8	4	14	3	18	26	36	18	60	42	234
Divisions	1	4	4	4	7	2	3	13	6	5	10	4	63
Autjugacy classes	1	3	2	3	4	2	2	5	4	3	5	3	37

Dimension	1	2	3	4	6	8	12	16	18	36	48
Irr. complex chars.	108	27	36	54	9	0	0	0	0	0	0	234
Irr. rational chars.	4	17	4	4	17	1	4	4	4	1	3	63

Minimal presentations

Permutation degree:	$57$
Transitive degree:	$432$
Rank:	$2$
Inequivalent generating pairs:	$36$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	48
Arbitrary	4	6	14

Constructions

Presentation:	$\langle a, b, c, d \mid b^{2}=c^{2}=d^{72}=[b,d]=[c,d]=1, a^{6}=d^{36}, b^{a}=c, c^{a}=bcd^{18}, d^{a}=d^{55}, c^{b}=cd^{36} \rangle$
Permutation group:	Degree $57$ $\langle(1,2,7,6)(3,12,17,28)(4,9,22,20)(5,11,23,10)(8,15,16,14)(13,29,33,42)(18,25,37,38) \!\cdots\! \rangle$
Direct product:	$C_9$ $\, \times\, $ $(Q_{16}.A_4)$
Semidirect product:	$(C_{72}.A_4)$ $\,\rtimes\,$ $C_2$	$(C_8.A_4)$ $\,\rtimes\,$ $C_{18}$ (3)	$(C_{72}.C_2^3)$ $\,\rtimes\,$ $C_3$	$(C_{24}.(C_2\times A_4))$ $\,\rtimes\,$ $C_3$ (2)	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_9\times Q_{16})$ . $A_4$	$Q_{16}$ . $(C_9\times A_4)$	$Q_8$ . $(A_4\times C_{18})$ (2)	$C_{18}$ . $(D_4\times A_4)$	all 42

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

Homology

Abelianization:	$C_{6} \times C_{18} \simeq C_{2}^{2} \times C_{3} \times C_{9}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 1089 subgroups in 287 conjugacy classes, 81 normal (35 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{18}$	$G/Z \simeq$ $D_4\times A_4$
Commutator:	$G' \simeq$ $D_4:C_2$	$G/G' \simeq$ $C_6\times C_{18}$
Frattini:	$\Phi \simeq$ $C_{12}$	$G/\Phi \simeq$ $C_2^2:C_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{72}.C_2^3$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_9\times Q_{16}.A_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_2^3.C_6^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$

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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Subgroup information

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Classes of subgroups up to conjugation

Order 1728: $C_9\times Q_{16}.A_4$

Order 864: $C_9\times \SL(2,3):C_2^2$ x 2, $C_{72}.A_4$

Order 576: $C_{24}.(C_2\times A_4)$ x 2, $C_3\times Q_{16}.A_4$, $C_{72}.C_2^3$

Order 432: $\SL(2,3):C_{18}$ x 3, $C_{24}.C_{18}$

Order 288: $C_{12}.(C_2\times A_4)$ x 4, $Q_8.C_6^2$ x 2, $C_{24}.A_4$ x 2, $C_{36}.C_2^3$ x 2, $D_8:C_{18}$ x 2, $C_{24}.A_4$, $D_8:C_{18}$, $D_8\times C_{18}$, $\OD_{16}:C_{18}$

Order 216: $C_{12}.C_{18}$ x 2, $C_9\times \SL(2,3)$, $C_3\times C_{72}$

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

Order 144: $C_{12}.A_4$ x 6, $D_4:C_{18}$ x 5, $D_4\times C_{18}$ x 4, $C_9\times Q_{16}$ x 3, $C_9\times D_8$ x 3, $\SL(2,3):C_6$ x 3, $C_9\times \SD_{16}$ x 2, $C_9\times \OD_{16}$, $C_2\times C_{72}$, $C_3^2\times Q_{16}$

Order 108: $C_3\times C_{36}$ x 3

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: $Q_8\times C_9$ x 7, $D_4\times C_9$ x 7, $C_{72}$ x 4, $C_2\times C_{36}$ x 3, $Q_8\times C_3^2$ x 2, $Q_8:C_9$ x 2, $C_2^2\times C_{18}$ x 2, $C_3\times \SL(2,3)$, $C_3\times C_{24}$

Order 64: $D_8:C_2^2$

Order 54: $C_3\times C_{18}$

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: $C_{36}$ x 10, $C_2\times C_{18}$ x 5, $C_3\times C_{12}$ 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 27: $C_3\times C_9$

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: $C_{18}$ x 6, $C_3\times C_6$

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: $C_9$ x 3, $C_3^2$

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 1728: $C_9\times Q_{16}.A_4$

Order 864: $C_9\times \SL(2,3):C_2^2$, $C_{72}.A_4$

Order 576: $C_3\times Q_{16}.A_4$, $C_{72}.C_2^3$, $C_{24}.(C_2\times A_4)$

Order 432: $\SL(2,3):C_{18}$ x 2, $C_{24}.C_{18}$

Order 288: $Q_8.C_6^2$, $C_{24}.A_4$, $C_{24}.A_4$, $C_{36}.C_2^3$, $C_{12}.(C_2\times A_4)$, $D_8:C_{18}$, $D_8:C_{18}$, $D_8\times C_{18}$, $\OD_{16}:C_{18}$

Order 216: $C_{12}.C_{18}$, $C_9\times \SL(2,3)$, $C_3\times C_{72}$

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

Order 144: $D_4:C_{18}$ x 3, $D_4\times C_{18}$ x 2, $C_{12}.A_4$ x 2, $C_9\times Q_{16}$ x 2, $C_9\times D_8$ x 2, $\SL(2,3):C_6$ x 2, $C_9\times \SD_{16}$, $C_9\times \OD_{16}$, $C_2\times C_{72}$, $C_3^2\times Q_{16}$

Order 108: $C_3\times C_{36}$ x 2

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: $D_4\times C_9$ x 4, $C_{72}$ x 3, $Q_8\times C_9$ x 3, $C_2\times C_{36}$ x 2, $Q_8\times C_3^2$, $Q_8:C_9$, $C_3\times \SL(2,3)$, $C_2^2\times C_{18}$, $C_3\times C_{24}$

Order 64: $D_8:C_2^2$

Order 54: $C_3\times C_{18}$

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: $C_{36}$ x 5, $C_2\times C_{18}$ x 3, $C_3\times C_{12}$ 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 27: $C_3\times C_9$

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: $C_{18}$ x 4, $C_3\times C_6$

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: $C_9$ x 2, $C_3^2$

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 1728: $C_9\times Q_{16}.A_4$ ($C_1$)

Order 864: $C_9\times \SL(2,3):C_2^2$ ($C_2$) x 2, $C_{72}.A_4$ ($C_2$)

Order 576: $C_{24}.(C_2\times A_4)$ ($C_3$) x 2, $C_3\times Q_{16}.A_4$ ($C_3$), $C_{72}.C_2^3$ ($C_3$)

Order 432: $\SL(2,3):C_{18}$ ($C_2^2$)

Order 288: $C_{12}.(C_2\times A_4)$ ($C_6$) x 4, $Q_8.C_6^2$ ($C_6$) x 2, $C_{24}.A_4$ ($C_6$) x 2, $C_{36}.C_2^3$ ($C_6$) x 2, $C_{24}.A_4$ ($C_6$), $\OD_{16}:C_{18}$ ($C_6$)

Order 216: $C_9\times \SL(2,3)$ ($D_4$)

Order 192: $Q_{16}.A_4$ ($C_9$) x 3, $C_{12}.C_2^4$ ($C_3^2$)

Order 144: $C_{12}.A_4$ ($C_2\times C_6$) x 2, $D_4:C_{18}$ ($C_2\times C_6$), $C_9\times Q_{16}$ ($A_4$), $\SL(2,3):C_6$ ($C_2\times C_6$)

Order 96: $\SL(2,3):C_2^2$ ($C_{18}$) x 6, $C_8.A_4$ ($C_{18}$) x 3, $C_6.C_2^4$ ($C_3\times C_6$) x 2, $\OD_{16}:C_6$ ($C_3\times C_6$)

Order 72: $Q_8:C_9$ ($C_3\times D_4$) x 2, $Q_8\times C_9$ ($C_2\times A_4$) x 2, $C_3\times \SL(2,3)$ ($C_3\times D_4$), $C_{72}$ ($C_2\times A_4$), $Q_8\times C_9$ ($C_3\times D_4$)

Order 64: $D_8:C_2^2$ ($C_3\times C_9$)

Order 48: $\SL(2,3):C_2$ ($C_2\times C_{18}$) x 3, $D_4:C_6$ ($C_6^2$), $C_3\times Q_{16}$ ($C_3\times A_4$)

Order 36: $C_{36}$ ($C_2^2\times A_4$)

Order 32: $D_4:C_2^2$ ($C_3\times C_{18}$) x 2, $\OD_{16}:C_2$ ($C_3\times C_{18}$)

Order 24: $\SL(2,3)$ ($D_4\times C_9$) x 3, $C_3\times Q_8$ ($C_6\times A_4$) x 2, $C_{24}$ ($C_6\times A_4$), $C_3\times Q_8$ ($D_4\times C_3^2$)

Order 18: $C_{18}$ ($D_4\times A_4$)

Order 16: $Q_{16}$ ($C_9\times A_4$), $D_4:C_2$ ($C_6\times C_{18}$)

Order 12: $C_{12}$ ($C_2^2:C_6^2$)

Order 9: $C_9$ ($Q_{16}.A_4$)

Order 8: $Q_8$ ($A_4\times C_{18}$) x 2, $Q_8$ ($C_{12}:C_{18}$), $C_8$ ($A_4\times C_{18}$)

Order 6: $C_6$ ($C_2^3.C_6^2$)

Order 4: $C_4$ ($C_2\times A_4\times C_{18}$)

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

Order 2: $C_2$ ($D_4\times C_9\times A_4$)

Order 1: $C_1$ ($C_9\times Q_{16}.A_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $C_9\times Q_{16}.A_4$ ($C_1$)

Order 864: $C_9\times \SL(2,3):C_2^2$ ($C_2$), $C_{72}.A_4$ ($C_2$)

Order 576: $C_3\times Q_{16}.A_4$ ($C_3$), $C_{72}.C_2^3$ ($C_3$), $C_{24}.(C_2\times A_4)$ ($C_3$)

Order 432: $\SL(2,3):C_{18}$ ($C_2^2$)

Order 288: $Q_8.C_6^2$ ($C_6$), $C_{24}.A_4$ ($C_6$), $C_{24}.A_4$ ($C_6$), $C_{36}.C_2^3$ ($C_6$), $C_{12}.(C_2\times A_4)$ ($C_6$), $\OD_{16}:C_{18}$ ($C_6$)

Order 216: $C_9\times \SL(2,3)$ ($D_4$)

Order 192: $C_{12}.C_2^4$ ($C_3^2$), $Q_{16}.A_4$ ($C_9$)

Order 144: $D_4:C_{18}$ ($C_2\times C_6$), $C_{12}.A_4$ ($C_2\times C_6$), $C_9\times Q_{16}$ ($A_4$), $\SL(2,3):C_6$ ($C_2\times C_6$)

Order 96: $C_8.A_4$ ($C_{18}$), $C_6.C_2^4$ ($C_3\times C_6$), $\SL(2,3):C_2^2$ ($C_{18}$), $\OD_{16}:C_6$ ($C_3\times C_6$)

Order 72: $Q_8:C_9$ ($C_3\times D_4$), $C_3\times \SL(2,3)$ ($C_3\times D_4$), $C_{72}$ ($C_2\times A_4$), $Q_8\times C_9$ ($C_2\times A_4$), $Q_8\times C_9$ ($C_3\times D_4$)

Order 64: $D_8:C_2^2$ ($C_3\times C_9$)

Order 48: $D_4:C_6$ ($C_6^2$), $\SL(2,3):C_2$ ($C_2\times C_{18}$), $C_3\times Q_{16}$ ($C_3\times A_4$)

Order 36: $C_{36}$ ($C_2^2\times A_4$)

Order 32: $D_4:C_2^2$ ($C_3\times C_{18}$), $\OD_{16}:C_2$ ($C_3\times C_{18}$)

Order 24: $\SL(2,3)$ ($D_4\times C_9$), $C_{24}$ ($C_6\times A_4$), $C_3\times Q_8$ ($C_6\times A_4$), $C_3\times Q_8$ ($D_4\times C_3^2$)

Order 18: $C_{18}$ ($D_4\times A_4$)

Order 16: $Q_{16}$ ($C_9\times A_4$), $D_4:C_2$ ($C_6\times C_{18}$)

Order 12: $C_{12}$ ($C_2^2:C_6^2$)

Order 9: $C_9$ ($Q_{16}.A_4$)

Order 8: $Q_8$ ($C_{12}:C_{18}$), $Q_8$ ($A_4\times C_{18}$), $C_8$ ($A_4\times C_{18}$)

Order 6: $C_6$ ($C_2^3.C_6^2$)

Order 4: $C_4$ ($C_2\times A_4\times C_{18}$)

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

Order 2: $C_2$ ($D_4\times C_9\times A_4$)

Order 1: $C_1$ ($C_9\times Q_{16}.A_4$)

Series

Derived series

$C_9\times Q_{16}.A_4$

$\rhd$

$D_4:C_2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_9\times Q_{16}.A_4$

$\rhd$

$C_{72}.A_4$

$\rhd$

$\OD_{16}:C_{18}$

$\rhd$

$C_{72}$

$\rhd$

$C_{36}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_9\times Q_{16}.A_4$

$\rhd$

$D_4:C_2$

$\rhd$

$Q_8$

Upper central series

$C_1$

$\lhd$

$C_{18}$

$\lhd$

$C_{36}$

$\lhd$

$C_9\times Q_{16}$

Character theory

Complex character table

See the $234 \times 234$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $63 \times 63$ rational character table.