Aliases: Dic9.2A4, SL2(𝔽3)⋊2D9, C6.2(S3×A4), C2.2(A4×D9), C9⋊3(C4.A4), C18.7(C2×A4), (Q8×C9).3C6, Q8.2(C3×D9), Q8⋊3D9⋊3C3, (C9×SL2(𝔽3))⋊2C2, C3.2(Dic3.A4), (C3×SL2(𝔽3)).6S3, (C3×Q8).12(C3×S3), SmallGroup(432,262)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8×C9 — Dic9.2A4 |
Generators and relations for Dic9.2A4
G = < a,b,c,d,e | a18=e3=1, b2=c2=d2=a9, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a9c, ece-1=a9cd, ede-1=c >
54C2
4C3
8C3
3C4
9C4
27C22
4C6
8C6
18S3
4C32
8C9
27C2×C4
27D4
3C12
3Dic3
9D6
36C12
6D9
8C18
9D12
3D18
3C36
12C3×Dic3
3D36
Smallest permutation representation of Dic9.2A4
►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 88 10 79)(2 87 11 78)(3 86 12 77)(4 85 13 76)(5 84 14 75)(6 83 15 74)(7 82 16 73)(8 81 17 90)(9 80 18 89)(19 107 28 98)(20 106 29 97)(21 105 30 96)(22 104 31 95)(23 103 32 94)(24 102 33 93)(25 101 34 92)(26 100 35 91)(27 99 36 108)(37 122 46 113)(38 121 47 112)(39 120 48 111)(40 119 49 110)(41 118 50 109)(42 117 51 126)(43 116 52 125)(44 115 53 124)(45 114 54 123)(55 144 64 135)(56 143 65 134)(57 142 66 133)(58 141 67 132)(59 140 68 131)(60 139 69 130)(61 138 70 129)(62 137 71 128)(63 136 72 127)
(1 35 10 26)(2 36 11 27)(3 19 12 28)(4 20 13 29)(5 21 14 30)(6 22 15 31)(7 23 16 32)(8 24 17 33)(9 25 18 34)(37 62 46 71)(38 63 47 72)(39 64 48 55)(40 65 49 56)(41 66 50 57)(42 67 51 58)(43 68 52 59)(44 69 53 60)(45 70 54 61)(73 94 82 103)(74 95 83 104)(75 96 84 105)(76 97 85 106)(77 98 86 107)(78 99 87 108)(79 100 88 91)(80 101 89 92)(81 102 90 93)(109 142 118 133)(110 143 119 134)(111 144 120 135)(112 127 121 136)(113 128 122 137)(114 129 123 138)(115 130 124 139)(116 131 125 140)(117 132 126 141)
(1 50 10 41)(2 51 11 42)(3 52 12 43)(4 53 13 44)(5 54 14 45)(6 37 15 46)(7 38 16 47)(8 39 17 48)(9 40 18 49)(19 68 28 59)(20 69 29 60)(21 70 30 61)(22 71 31 62)(23 72 32 63)(24 55 33 64)(25 56 34 65)(26 57 35 66)(27 58 36 67)(73 112 82 121)(74 113 83 122)(75 114 84 123)(76 115 85 124)(77 116 86 125)(78 117 87 126)(79 118 88 109)(80 119 89 110)(81 120 90 111)(91 133 100 142)(92 134 101 143)(93 135 102 144)(94 136 103 127)(95 137 104 128)(96 138 105 129)(97 139 106 130)(98 140 107 131)(99 141 108 132)
(19 52 59)(20 53 60)(21 54 61)(22 37 62)(23 38 63)(24 39 64)(25 40 65)(26 41 66)(27 42 67)(28 43 68)(29 44 69)(30 45 70)(31 46 71)(32 47 72)(33 48 55)(34 49 56)(35 50 57)(36 51 58)(91 109 142)(92 110 143)(93 111 144)(94 112 127)(95 113 128)(96 114 129)(97 115 130)(98 116 131)(99 117 132)(100 118 133)(101 119 134)(102 120 135)(103 121 136)(104 122 137)(105 123 138)(106 124 139)(107 125 140)(108 126 141)
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), (1,88,10,79)(2,87,11,78)(3,86,12,77)(4,85,13,76)(5,84,14,75)(6,83,15,74)(7,82,16,73)(8,81,17,90)(9,80,18,89)(19,107,28,98)(20,106,29,97)(21,105,30,96)(22,104,31,95)(23,103,32,94)(24,102,33,93)(25,101,34,92)(26,100,35,91)(27,99,36,108)(37,122,46,113)(38,121,47,112)(39,120,48,111)(40,119,49,110)(41,118,50,109)(42,117,51,126)(43,116,52,125)(44,115,53,124)(45,114,54,123)(55,144,64,135)(56,143,65,134)(57,142,66,133)(58,141,67,132)(59,140,68,131)(60,139,69,130)(61,138,70,129)(62,137,71,128)(63,136,72,127), (1,35,10,26)(2,36,11,27)(3,19,12,28)(4,20,13,29)(5,21,14,30)(6,22,15,31)(7,23,16,32)(8,24,17,33)(9,25,18,34)(37,62,46,71)(38,63,47,72)(39,64,48,55)(40,65,49,56)(41,66,50,57)(42,67,51,58)(43,68,52,59)(44,69,53,60)(45,70,54,61)(73,94,82,103)(74,95,83,104)(75,96,84,105)(76,97,85,106)(77,98,86,107)(78,99,87,108)(79,100,88,91)(80,101,89,92)(81,102,90,93)(109,142,118,133)(110,143,119,134)(111,144,120,135)(112,127,121,136)(113,128,122,137)(114,129,123,138)(115,130,124,139)(116,131,125,140)(117,132,126,141), (1,50,10,41)(2,51,11,42)(3,52,12,43)(4,53,13,44)(5,54,14,45)(6,37,15,46)(7,38,16,47)(8,39,17,48)(9,40,18,49)(19,68,28,59)(20,69,29,60)(21,70,30,61)(22,71,31,62)(23,72,32,63)(24,55,33,64)(25,56,34,65)(26,57,35,66)(27,58,36,67)(73,112,82,121)(74,113,83,122)(75,114,84,123)(76,115,85,124)(77,116,86,125)(78,117,87,126)(79,118,88,109)(80,119,89,110)(81,120,90,111)(91,133,100,142)(92,134,101,143)(93,135,102,144)(94,136,103,127)(95,137,104,128)(96,138,105,129)(97,139,106,130)(98,140,107,131)(99,141,108,132), (19,52,59)(20,53,60)(21,54,61)(22,37,62)(23,38,63)(24,39,64)(25,40,65)(26,41,66)(27,42,67)(28,43,68)(29,44,69)(30,45,70)(31,46,71)(32,47,72)(33,48,55)(34,49,56)(35,50,57)(36,51,58)(91,109,142)(92,110,143)(93,111,144)(94,112,127)(95,113,128)(96,114,129)(97,115,130)(98,116,131)(99,117,132)(100,118,133)(101,119,134)(102,120,135)(103,121,136)(104,122,137)(105,123,138)(106,124,139)(107,125,140)(108,126,141)>;
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), (1,88,10,79)(2,87,11,78)(3,86,12,77)(4,85,13,76)(5,84,14,75)(6,83,15,74)(7,82,16,73)(8,81,17,90)(9,80,18,89)(19,107,28,98)(20,106,29,97)(21,105,30,96)(22,104,31,95)(23,103,32,94)(24,102,33,93)(25,101,34,92)(26,100,35,91)(27,99,36,108)(37,122,46,113)(38,121,47,112)(39,120,48,111)(40,119,49,110)(41,118,50,109)(42,117,51,126)(43,116,52,125)(44,115,53,124)(45,114,54,123)(55,144,64,135)(56,143,65,134)(57,142,66,133)(58,141,67,132)(59,140,68,131)(60,139,69,130)(61,138,70,129)(62,137,71,128)(63,136,72,127), (1,35,10,26)(2,36,11,27)(3,19,12,28)(4,20,13,29)(5,21,14,30)(6,22,15,31)(7,23,16,32)(8,24,17,33)(9,25,18,34)(37,62,46,71)(38,63,47,72)(39,64,48,55)(40,65,49,56)(41,66,50,57)(42,67,51,58)(43,68,52,59)(44,69,53,60)(45,70,54,61)(73,94,82,103)(74,95,83,104)(75,96,84,105)(76,97,85,106)(77,98,86,107)(78,99,87,108)(79,100,88,91)(80,101,89,92)(81,102,90,93)(109,142,118,133)(110,143,119,134)(111,144,120,135)(112,127,121,136)(113,128,122,137)(114,129,123,138)(115,130,124,139)(116,131,125,140)(117,132,126,141), (1,50,10,41)(2,51,11,42)(3,52,12,43)(4,53,13,44)(5,54,14,45)(6,37,15,46)(7,38,16,47)(8,39,17,48)(9,40,18,49)(19,68,28,59)(20,69,29,60)(21,70,30,61)(22,71,31,62)(23,72,32,63)(24,55,33,64)(25,56,34,65)(26,57,35,66)(27,58,36,67)(73,112,82,121)(74,113,83,122)(75,114,84,123)(76,115,85,124)(77,116,86,125)(78,117,87,126)(79,118,88,109)(80,119,89,110)(81,120,90,111)(91,133,100,142)(92,134,101,143)(93,135,102,144)(94,136,103,127)(95,137,104,128)(96,138,105,129)(97,139,106,130)(98,140,107,131)(99,141,108,132), (19,52,59)(20,53,60)(21,54,61)(22,37,62)(23,38,63)(24,39,64)(25,40,65)(26,41,66)(27,42,67)(28,43,68)(29,44,69)(30,45,70)(31,46,71)(32,47,72)(33,48,55)(34,49,56)(35,50,57)(36,51,58)(91,109,142)(92,110,143)(93,111,144)(94,112,127)(95,113,128)(96,114,129)(97,115,130)(98,116,131)(99,117,132)(100,118,133)(101,119,134)(102,120,135)(103,121,136)(104,122,137)(105,123,138)(106,124,139)(107,125,140)(108,126,141) );
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)], [(1,88,10,79),(2,87,11,78),(3,86,12,77),(4,85,13,76),(5,84,14,75),(6,83,15,74),(7,82,16,73),(8,81,17,90),(9,80,18,89),(19,107,28,98),(20,106,29,97),(21,105,30,96),(22,104,31,95),(23,103,32,94),(24,102,33,93),(25,101,34,92),(26,100,35,91),(27,99,36,108),(37,122,46,113),(38,121,47,112),(39,120,48,111),(40,119,49,110),(41,118,50,109),(42,117,51,126),(43,116,52,125),(44,115,53,124),(45,114,54,123),(55,144,64,135),(56,143,65,134),(57,142,66,133),(58,141,67,132),(59,140,68,131),(60,139,69,130),(61,138,70,129),(62,137,71,128),(63,136,72,127)], [(1,35,10,26),(2,36,11,27),(3,19,12,28),(4,20,13,29),(5,21,14,30),(6,22,15,31),(7,23,16,32),(8,24,17,33),(9,25,18,34),(37,62,46,71),(38,63,47,72),(39,64,48,55),(40,65,49,56),(41,66,50,57),(42,67,51,58),(43,68,52,59),(44,69,53,60),(45,70,54,61),(73,94,82,103),(74,95,83,104),(75,96,84,105),(76,97,85,106),(77,98,86,107),(78,99,87,108),(79,100,88,91),(80,101,89,92),(81,102,90,93),(109,142,118,133),(110,143,119,134),(111,144,120,135),(112,127,121,136),(113,128,122,137),(114,129,123,138),(115,130,124,139),(116,131,125,140),(117,132,126,141)], [(1,50,10,41),(2,51,11,42),(3,52,12,43),(4,53,13,44),(5,54,14,45),(6,37,15,46),(7,38,16,47),(8,39,17,48),(9,40,18,49),(19,68,28,59),(20,69,29,60),(21,70,30,61),(22,71,31,62),(23,72,32,63),(24,55,33,64),(25,56,34,65),(26,57,35,66),(27,58,36,67),(73,112,82,121),(74,113,83,122),(75,114,84,123),(76,115,85,124),(77,116,86,125),(78,117,87,126),(79,118,88,109),(80,119,89,110),(81,120,90,111),(91,133,100,142),(92,134,101,143),(93,135,102,144),(94,136,103,127),(95,137,104,128),(96,138,105,129),(97,139,106,130),(98,140,107,131),(99,141,108,132)], [(19,52,59),(20,53,60),(21,54,61),(22,37,62),(23,38,63),(24,39,64),(25,40,65),(26,41,66),(27,42,67),(28,43,68),(29,44,69),(30,45,70),(31,46,71),(32,47,72),(33,48,55),(34,49,56),(35,50,57),(36,51,58),(91,109,142),(92,110,143),(93,111,144),(94,112,127),(95,113,128),(96,114,129),(97,115,130),(98,116,131),(99,117,132),(100,118,133),(101,119,134),(102,120,135),(103,121,136),(104,122,137),(105,123,138),(106,124,139),(107,125,140),(108,126,141)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 9D | ··· | 9I | 12A | 12B | 12C | 12D | 12E | 18A | 18B | 18C | 18D | ··· | 18I | 36A | 36B | 36C |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | 36 | 36 |
| size | 1 | 1 | 54 | 2 | 4 | 4 | 8 | 8 | 6 | 9 | 9 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 8 | ··· | 8 | 12 | 36 | 36 | 36 | 36 | 2 | 2 | 2 | 8 | ··· | 8 | 12 | 12 | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C3 | C6 | S3 | D9 | C3×S3 | C4.A4 | C3×D9 | A4 | C2×A4 | Dic3.A4 | Dic3.A4 | Dic9.2A4 | Dic9.2A4 | S3×A4 | A4×D9 |
| kernel | Dic9.2A4 | C9×SL2(𝔽3) | Q8⋊3D9 | Q8×C9 | C3×SL2(𝔽3) | SL2(𝔽3) | C3×Q8 | C9 | Q8 | Dic9 | C18 | C3 | C3 | C1 | C1 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 3 | 2 | 6 | 6 | 1 | 1 | 1 | 2 | 3 | 6 | 1 | 3 |
Matrix representation of Dic9.2A4 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 33 | 0 |
| 0 | 0 | 3 | 9 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 5 | 34 |
| 0 | 0 | 8 | 32 |
| 0 | 1 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 10 | 0 | 0 |
| 10 | 26 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 10 | 26 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,33,3,0,0,0,9],[6,0,0,0,0,6,0,0,0,0,5,8,0,0,34,32],[0,36,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[11,10,0,0,10,26,0,0,0,0,1,0,0,0,0,1],[1,10,0,0,0,26,0,0,0,0,1,0,0,0,0,1] >;
Dic9.2A4 in GAP, Magma, Sage, TeX
{\rm Dic}_9._2A_4 % in TeX
G:=Group("Dic9.2A4"); // GroupNames label
G:=SmallGroup(432,262);
// by ID
G=gap.SmallGroup(432,262);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-3,-2,-3,504,198,268,94,409,192,6724,452,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^18=e^3=1,b^2=c^2=d^2=a^9,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^9*c,e*c*e^-1=a^9*c*d,e*d*e^-1=c>;
// generators/relations
Export