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G = C9⋊Q16order 144 = 24·32

The semidirect product of C9 and Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C92Q16, C18.9D4, C4.3D18, C12.3D6, Q8.2D9, C36.3C22, Dic18.2C2, C9⋊C8.C2, C3.(C3⋊Q16), (C3×Q8).5S3, (Q8×C9).1C2, C2.6(C9⋊D4), C6.16(C3⋊D4), SmallGroup(144,17)

Series: Derived Chief Lower central Upper central

C1C36 — C9⋊Q16
C1C3C9C18C36Dic18 — C9⋊Q16
C9C18C36 — C9⋊Q16
C1C2C4Q8

Generators and relations for C9⋊Q16
 G = < a,b,c | a9=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >

2C4
18C4
9C8
9Q8
2C12
6Dic3
9Q16
3C3⋊C8
3Dic6
2Dic9
2C36
3C3⋊Q16

Character table of C9⋊Q16

 class 1234A4B4C68A8B9A9B9C12A12B12C18A18B18C36A36B36C36D36E36F36G36H36I
 size 112243621818222444222444444444
ρ1111111111111111111111111111    trivial
ρ211111-11-1-1111111111111111111    linear of order 2
ρ31111-111-1-11111-1-1111-1-1-1-1-1-1111    linear of order 2
ρ41111-1-11111111-1-1111-1-1-1-1-1-1111    linear of order 2
ρ5222220200-1-1-1222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62222-20200-1-1-12-2-2-1-1-1111111-1-1-1    orthogonal lifted from D6
ρ7222-200200222-200222000000-2-2-2    orthogonal lifted from D4
ρ822-12-20-100ζ9792ζ9594ζ989-111ζ9594ζ989ζ97929899792959497929899594ζ989ζ9792ζ9594    orthogonal lifted from D18
ρ922-1220-100ζ9792ζ9594ζ989-1-1-1ζ9594ζ989ζ9792ζ989ζ9792ζ9594ζ9792ζ989ζ9594ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ1022-12-20-100ζ989ζ9792ζ9594-111ζ9792ζ9594ζ9899594989979298995949792ζ9594ζ989ζ9792    orthogonal lifted from D18
ρ1122-1220-100ζ989ζ9792ζ9594-1-1-1ζ9792ζ9594ζ989ζ9594ζ989ζ9792ζ989ζ9594ζ9792ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ1222-1220-100ζ9594ζ989ζ9792-1-1-1ζ989ζ9792ζ9594ζ9792ζ9594ζ989ζ9594ζ9792ζ989ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ1322-12-20-100ζ9594ζ989ζ9792-111ζ989ζ9792ζ95949792959498995949792989ζ9792ζ9594ζ989    orthogonal lifted from D18
ρ142-22000-22-2222000-2-2-2000000000    symplectic lifted from Q16, Schur index 2
ρ152-22000-2-22222000-2-2-2000000000    symplectic lifted from Q16, Schur index 2
ρ16222-200200-1-1-1-200-1-1-1--3--3--3-3-3-3111    complex lifted from C3⋊D4
ρ1722-1-200-100ζ9792ζ9594ζ9891--3-3ζ9594ζ989ζ9792989ζ979295949792ζ989ζ959498997929594    complex lifted from C9⋊D4
ρ1822-1-200-100ζ9594ζ989ζ97921-3--3ζ989ζ9792ζ95949792ζ9594ζ9899594ζ979298997929594989    complex lifted from C9⋊D4
ρ1922-1-200-100ζ9594ζ989ζ97921--3-3ζ989ζ9792ζ9594ζ97929594989ζ95949792ζ98997929594989    complex lifted from C9⋊D4
ρ20222-200200-1-1-1-200-1-1-1-3-3-3--3--3--3111    complex lifted from C3⋊D4
ρ2122-1-200-100ζ989ζ9792ζ95941-3--3ζ9792ζ9594ζ989ζ9594ζ98997929899594ζ979295949899792    complex lifted from C9⋊D4
ρ2222-1-200-100ζ9792ζ9594ζ9891-3--3ζ9594ζ989ζ9792ζ9899792ζ9594ζ9792989959498997929594    complex lifted from C9⋊D4
ρ2322-1-200-100ζ989ζ9792ζ95941--3-3ζ9792ζ9594ζ9899594989ζ9792ζ989ζ9594979295949899792    complex lifted from C9⋊D4
ρ244-44000-400-2-2-2000222000000000    symplectic lifted from C3⋊Q16, Schur index 2
ρ254-4-200020098+2ζ997+2ζ9295+2ζ94000-2ζ97-2ζ92-2ζ95-2ζ94-2ζ98-2ζ9000000000    symplectic faithful, Schur index 2
ρ264-4-200020097+2ζ9295+2ζ9498+2ζ9000-2ζ95-2ζ94-2ζ98-2ζ9-2ζ97-2ζ92000000000    symplectic faithful, Schur index 2
ρ274-4-200020095+2ζ9498+2ζ997+2ζ92000-2ζ98-2ζ9-2ζ97-2ζ92-2ζ95-2ζ94000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C9⋊Q16
Regular action on 144 points
Generators in S144
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117)(118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135)(136 137 138 139 140 141 142 143 144)
(1 133 20 124 11 142 29 115)(2 132 21 123 12 141 30 114)(3 131 22 122 13 140 31 113)(4 130 23 121 14 139 32 112)(5 129 24 120 15 138 33 111)(6 128 25 119 16 137 34 110)(7 127 26 118 17 136 35 109)(8 135 27 126 18 144 36 117)(9 134 19 125 10 143 28 116)(37 98 64 80 46 107 55 89)(38 97 65 79 47 106 56 88)(39 96 66 78 48 105 57 87)(40 95 67 77 49 104 58 86)(41 94 68 76 50 103 59 85)(42 93 69 75 51 102 60 84)(43 92 70 74 52 101 61 83)(44 91 71 73 53 100 62 82)(45 99 72 81 54 108 63 90)
(1 47 11 38)(2 48 12 39)(3 49 13 40)(4 50 14 41)(5 51 15 42)(6 52 16 43)(7 53 17 44)(8 54 18 45)(9 46 10 37)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)(73 136 82 127)(74 137 83 128)(75 138 84 129)(76 139 85 130)(77 140 86 131)(78 141 87 132)(79 142 88 133)(80 143 89 134)(81 144 90 135)(91 109 100 118)(92 110 101 119)(93 111 102 120)(94 112 103 121)(95 113 104 122)(96 114 105 123)(97 115 106 124)(98 116 107 125)(99 117 108 126)

G:=sub<Sym(144)| (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144), (1,133,20,124,11,142,29,115)(2,132,21,123,12,141,30,114)(3,131,22,122,13,140,31,113)(4,130,23,121,14,139,32,112)(5,129,24,120,15,138,33,111)(6,128,25,119,16,137,34,110)(7,127,26,118,17,136,35,109)(8,135,27,126,18,144,36,117)(9,134,19,125,10,143,28,116)(37,98,64,80,46,107,55,89)(38,97,65,79,47,106,56,88)(39,96,66,78,48,105,57,87)(40,95,67,77,49,104,58,86)(41,94,68,76,50,103,59,85)(42,93,69,75,51,102,60,84)(43,92,70,74,52,101,61,83)(44,91,71,73,53,100,62,82)(45,99,72,81,54,108,63,90), (1,47,11,38)(2,48,12,39)(3,49,13,40)(4,50,14,41)(5,51,15,42)(6,52,16,43)(7,53,17,44)(8,54,18,45)(9,46,10,37)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63)(73,136,82,127)(74,137,83,128)(75,138,84,129)(76,139,85,130)(77,140,86,131)(78,141,87,132)(79,142,88,133)(80,143,89,134)(81,144,90,135)(91,109,100,118)(92,110,101,119)(93,111,102,120)(94,112,103,121)(95,113,104,122)(96,114,105,123)(97,115,106,124)(98,116,107,125)(99,117,108,126)>;

G:=Group( (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144), (1,133,20,124,11,142,29,115)(2,132,21,123,12,141,30,114)(3,131,22,122,13,140,31,113)(4,130,23,121,14,139,32,112)(5,129,24,120,15,138,33,111)(6,128,25,119,16,137,34,110)(7,127,26,118,17,136,35,109)(8,135,27,126,18,144,36,117)(9,134,19,125,10,143,28,116)(37,98,64,80,46,107,55,89)(38,97,65,79,47,106,56,88)(39,96,66,78,48,105,57,87)(40,95,67,77,49,104,58,86)(41,94,68,76,50,103,59,85)(42,93,69,75,51,102,60,84)(43,92,70,74,52,101,61,83)(44,91,71,73,53,100,62,82)(45,99,72,81,54,108,63,90), (1,47,11,38)(2,48,12,39)(3,49,13,40)(4,50,14,41)(5,51,15,42)(6,52,16,43)(7,53,17,44)(8,54,18,45)(9,46,10,37)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63)(73,136,82,127)(74,137,83,128)(75,138,84,129)(76,139,85,130)(77,140,86,131)(78,141,87,132)(79,142,88,133)(80,143,89,134)(81,144,90,135)(91,109,100,118)(92,110,101,119)(93,111,102,120)(94,112,103,121)(95,113,104,122)(96,114,105,123)(97,115,106,124)(98,116,107,125)(99,117,108,126) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117),(118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135),(136,137,138,139,140,141,142,143,144)], [(1,133,20,124,11,142,29,115),(2,132,21,123,12,141,30,114),(3,131,22,122,13,140,31,113),(4,130,23,121,14,139,32,112),(5,129,24,120,15,138,33,111),(6,128,25,119,16,137,34,110),(7,127,26,118,17,136,35,109),(8,135,27,126,18,144,36,117),(9,134,19,125,10,143,28,116),(37,98,64,80,46,107,55,89),(38,97,65,79,47,106,56,88),(39,96,66,78,48,105,57,87),(40,95,67,77,49,104,58,86),(41,94,68,76,50,103,59,85),(42,93,69,75,51,102,60,84),(43,92,70,74,52,101,61,83),(44,91,71,73,53,100,62,82),(45,99,72,81,54,108,63,90)], [(1,47,11,38),(2,48,12,39),(3,49,13,40),(4,50,14,41),(5,51,15,42),(6,52,16,43),(7,53,17,44),(8,54,18,45),(9,46,10,37),(19,64,28,55),(20,65,29,56),(21,66,30,57),(22,67,31,58),(23,68,32,59),(24,69,33,60),(25,70,34,61),(26,71,35,62),(27,72,36,63),(73,136,82,127),(74,137,83,128),(75,138,84,129),(76,139,85,130),(77,140,86,131),(78,141,87,132),(79,142,88,133),(80,143,89,134),(81,144,90,135),(91,109,100,118),(92,110,101,119),(93,111,102,120),(94,112,103,121),(95,113,104,122),(96,114,105,123),(97,115,106,124),(98,116,107,125),(99,117,108,126)])

C9⋊Q16 is a maximal subgroup of
SD16⋊D9  SD163D9  Q16×D9  Q16⋊D9  C36.C23  D4.D18  D4.9D18  C27⋊Q16  Q8.D27  C12.D18  C9⋊Dic12  Dic18.C6  C36.19D6  C18.5S4
C9⋊Q16 is a maximal quotient of
C36.Q8  C18.Q16  Q82Dic9  C27⋊Q16  C12.D18  C9⋊Dic12  C36.19D6

Matrix representation of C9⋊Q16 in GL4(𝔽73) generated by

424500
287000
0010
0001
,
685000
55500
00048
003841
,
72000
07200
005433
002019
G:=sub<GL(4,GF(73))| [42,28,0,0,45,70,0,0,0,0,1,0,0,0,0,1],[68,55,0,0,50,5,0,0,0,0,0,38,0,0,48,41],[72,0,0,0,0,72,0,0,0,0,54,20,0,0,33,19] >;

C9⋊Q16 in GAP, Magma, Sage, TeX

C_9\rtimes Q_{16}
% in TeX

G:=Group("C9:Q16");
// GroupNames label

G:=SmallGroup(144,17);
// by ID

G=gap.SmallGroup(144,17);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,55,218,116,50,2404,208,3461]);
// Polycyclic

G:=Group<a,b,c|a^9=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C9⋊Q16 in TeX
Character table of C9⋊Q16 in TeX

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