metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic9⋊4D4, C23.19D18, C9⋊D4⋊C4, C9⋊2(C4×D4), C2.2(D4×D9), D18⋊C4⋊9C2, D18⋊2(C2×C4), C22⋊C4⋊7D9, C6.78(S3×D4), C22⋊2(C4×D9), Dic9⋊C4⋊9C2, Dic9⋊1(C2×C4), C18.18(C2×D4), (C2×C4).25D18, (C4×Dic9)⋊11C2, (C2×C12).177D6, C18.7(C22×C4), (C22×C6).41D6, C18.22(C4○D4), C2.2(D4⋊2D9), C3.(Dic3⋊4D4), (C2×C18).22C23, (C2×C36).54C22, (C22×Dic9)⋊1C2, C6.75(D4⋊2S3), C22.14(C22×D9), (C22×C18).11C22, (C2×Dic9).26C22, (C22×D9).16C22, (C2×C4×D9)⋊9C2, C2.9(C2×C4×D9), C6.46(S3×C2×C4), (C2×C18)⋊2(C2×C4), (C2×C6).7(C4×S3), (C9×C22⋊C4)⋊9C2, (C2×C9⋊D4).2C2, (C3×C22⋊C4).13S3, (C2×C6).179(C22×S3), SmallGroup(288,91)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic9⋊4D4
G = < a,b,c,d | a18=c4=d2=1, b2=a9, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 576 in 141 conjugacy classes, 54 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, C9, Dic3 [×5], C12 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4 [×2], C2×D4, D9 [×2], C18 [×3], C18 [×2], C4×S3 [×2], C2×Dic3 [×5], C3⋊D4 [×4], C2×C12 [×2], C22×S3, C22×C6, C4×D4, Dic9 [×4], Dic9, C36 [×2], D18 [×2], D18 [×2], C2×C18, C2×C18 [×2], C2×C18 [×2], C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C4×D9 [×2], C2×Dic9 [×3], C2×Dic9 [×2], C9⋊D4 [×4], C2×C36 [×2], C22×D9, C22×C18, Dic3⋊4D4, C4×Dic9, Dic9⋊C4, D18⋊C4, C9×C22⋊C4, C2×C4×D9, C22×Dic9, C2×C9⋊D4, Dic9⋊4D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, D9, C4×S3 [×2], C22×S3, C4×D4, D18 [×3], S3×C2×C4, S3×D4, D4⋊2S3, C4×D9 [×2], C22×D9, Dic3⋊4D4, C2×C4×D9, D4×D9, D4⋊2D9, Dic9⋊4D4
►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 69 10 60)(2 68 11 59)(3 67 12 58)(4 66 13 57)(5 65 14 56)(6 64 15 55)(7 63 16 72)(8 62 17 71)(9 61 18 70)(19 144 28 135)(20 143 29 134)(21 142 30 133)(22 141 31 132)(23 140 32 131)(24 139 33 130)(25 138 34 129)(26 137 35 128)(27 136 36 127)(37 117 46 126)(38 116 47 125)(39 115 48 124)(40 114 49 123)(41 113 50 122)(42 112 51 121)(43 111 52 120)(44 110 53 119)(45 109 54 118)(73 98 82 107)(74 97 83 106)(75 96 84 105)(76 95 85 104)(77 94 86 103)(78 93 87 102)(79 92 88 101)(80 91 89 100)(81 108 90 99)
(1 49 35 81)(2 48 36 80)(3 47 19 79)(4 46 20 78)(5 45 21 77)(6 44 22 76)(7 43 23 75)(8 42 24 74)(9 41 25 73)(10 40 26 90)(11 39 27 89)(12 38 28 88)(13 37 29 87)(14 54 30 86)(15 53 31 85)(16 52 32 84)(17 51 33 83)(18 50 34 82)(55 119 132 104)(56 118 133 103)(57 117 134 102)(58 116 135 101)(59 115 136 100)(60 114 137 99)(61 113 138 98)(62 112 139 97)(63 111 140 96)(64 110 141 95)(65 109 142 94)(66 126 143 93)(67 125 144 92)(68 124 127 91)(69 123 128 108)(70 122 129 107)(71 121 130 106)(72 120 131 105)
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 51)(20 52)(21 53)(22 54)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(61 100)(62 101)(63 102)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 91)(71 92)(72 93)(109 132)(110 133)(111 134)(112 135)(113 136)(114 137)(115 138)(116 139)(117 140)(118 141)(119 142)(120 143)(121 144)(122 127)(123 128)(124 129)(125 130)(126 131)
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), (1,69,10,60)(2,68,11,59)(3,67,12,58)(4,66,13,57)(5,65,14,56)(6,64,15,55)(7,63,16,72)(8,62,17,71)(9,61,18,70)(19,144,28,135)(20,143,29,134)(21,142,30,133)(22,141,31,132)(23,140,32,131)(24,139,33,130)(25,138,34,129)(26,137,35,128)(27,136,36,127)(37,117,46,126)(38,116,47,125)(39,115,48,124)(40,114,49,123)(41,113,50,122)(42,112,51,121)(43,111,52,120)(44,110,53,119)(45,109,54,118)(73,98,82,107)(74,97,83,106)(75,96,84,105)(76,95,85,104)(77,94,86,103)(78,93,87,102)(79,92,88,101)(80,91,89,100)(81,108,90,99), (1,49,35,81)(2,48,36,80)(3,47,19,79)(4,46,20,78)(5,45,21,77)(6,44,22,76)(7,43,23,75)(8,42,24,74)(9,41,25,73)(10,40,26,90)(11,39,27,89)(12,38,28,88)(13,37,29,87)(14,54,30,86)(15,53,31,85)(16,52,32,84)(17,51,33,83)(18,50,34,82)(55,119,132,104)(56,118,133,103)(57,117,134,102)(58,116,135,101)(59,115,136,100)(60,114,137,99)(61,113,138,98)(62,112,139,97)(63,111,140,96)(64,110,141,95)(65,109,142,94)(66,126,143,93)(67,125,144,92)(68,124,127,91)(69,123,128,108)(70,122,129,107)(71,121,130,106)(72,120,131,105), (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,51)(20,52)(21,53)(22,54)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,91)(71,92)(72,93)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,141)(119,142)(120,143)(121,144)(122,127)(123,128)(124,129)(125,130)(126,131)>;
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), (1,69,10,60)(2,68,11,59)(3,67,12,58)(4,66,13,57)(5,65,14,56)(6,64,15,55)(7,63,16,72)(8,62,17,71)(9,61,18,70)(19,144,28,135)(20,143,29,134)(21,142,30,133)(22,141,31,132)(23,140,32,131)(24,139,33,130)(25,138,34,129)(26,137,35,128)(27,136,36,127)(37,117,46,126)(38,116,47,125)(39,115,48,124)(40,114,49,123)(41,113,50,122)(42,112,51,121)(43,111,52,120)(44,110,53,119)(45,109,54,118)(73,98,82,107)(74,97,83,106)(75,96,84,105)(76,95,85,104)(77,94,86,103)(78,93,87,102)(79,92,88,101)(80,91,89,100)(81,108,90,99), (1,49,35,81)(2,48,36,80)(3,47,19,79)(4,46,20,78)(5,45,21,77)(6,44,22,76)(7,43,23,75)(8,42,24,74)(9,41,25,73)(10,40,26,90)(11,39,27,89)(12,38,28,88)(13,37,29,87)(14,54,30,86)(15,53,31,85)(16,52,32,84)(17,51,33,83)(18,50,34,82)(55,119,132,104)(56,118,133,103)(57,117,134,102)(58,116,135,101)(59,115,136,100)(60,114,137,99)(61,113,138,98)(62,112,139,97)(63,111,140,96)(64,110,141,95)(65,109,142,94)(66,126,143,93)(67,125,144,92)(68,124,127,91)(69,123,128,108)(70,122,129,107)(71,121,130,106)(72,120,131,105), (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,51)(20,52)(21,53)(22,54)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,91)(71,92)(72,93)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,141)(119,142)(120,143)(121,144)(122,127)(123,128)(124,129)(125,130)(126,131) );
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)], [(1,69,10,60),(2,68,11,59),(3,67,12,58),(4,66,13,57),(5,65,14,56),(6,64,15,55),(7,63,16,72),(8,62,17,71),(9,61,18,70),(19,144,28,135),(20,143,29,134),(21,142,30,133),(22,141,31,132),(23,140,32,131),(24,139,33,130),(25,138,34,129),(26,137,35,128),(27,136,36,127),(37,117,46,126),(38,116,47,125),(39,115,48,124),(40,114,49,123),(41,113,50,122),(42,112,51,121),(43,111,52,120),(44,110,53,119),(45,109,54,118),(73,98,82,107),(74,97,83,106),(75,96,84,105),(76,95,85,104),(77,94,86,103),(78,93,87,102),(79,92,88,101),(80,91,89,100),(81,108,90,99)], [(1,49,35,81),(2,48,36,80),(3,47,19,79),(4,46,20,78),(5,45,21,77),(6,44,22,76),(7,43,23,75),(8,42,24,74),(9,41,25,73),(10,40,26,90),(11,39,27,89),(12,38,28,88),(13,37,29,87),(14,54,30,86),(15,53,31,85),(16,52,32,84),(17,51,33,83),(18,50,34,82),(55,119,132,104),(56,118,133,103),(57,117,134,102),(58,116,135,101),(59,115,136,100),(60,114,137,99),(61,113,138,98),(62,112,139,97),(63,111,140,96),(64,110,141,95),(65,109,142,94),(66,126,143,93),(67,125,144,92),(68,124,127,91),(69,123,128,108),(70,122,129,107),(71,121,130,106),(72,120,131,105)], [(1,81),(2,82),(3,83),(4,84),(5,85),(6,86),(7,87),(8,88),(9,89),(10,90),(11,73),(12,74),(13,75),(14,76),(15,77),(16,78),(17,79),(18,80),(19,51),(20,52),(21,53),(22,54),(23,37),(24,38),(25,39),(26,40),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(55,94),(56,95),(57,96),(58,97),(59,98),(60,99),(61,100),(62,101),(63,102),(64,103),(65,104),(66,105),(67,106),(68,107),(69,108),(70,91),(71,92),(72,93),(109,132),(110,133),(111,134),(112,135),(113,136),(114,137),(115,138),(116,139),(117,140),(118,141),(119,142),(120,143),(121,144),(122,127),(123,128),(124,129),(125,130),(126,131)])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18O | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4○D4 | D9 | C4×S3 | D18 | D18 | C4×D9 | S3×D4 | D4⋊2S3 | D4×D9 | D4⋊2D9 |
| kernel | Dic9⋊4D4 | C4×Dic9 | Dic9⋊C4 | D18⋊C4 | C9×C22⋊C4 | C2×C4×D9 | C22×Dic9 | C2×C9⋊D4 | C9⋊D4 | C3×C22⋊C4 | Dic9 | C2×C12 | C22×C6 | C18 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 2 | 3 | 4 | 6 | 3 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of Dic9⋊4D4 ►in GL4(𝔽37) generated by
| 11 | 31 | 0 | 0 |
| 6 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 1 | 0 | 0 |
| 29 | 28 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 20 | 31 | 0 | 0 |
| 11 | 17 | 0 | 0 |
| 0 | 0 | 22 | 3 |
| 0 | 0 | 11 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 22 | 3 |
| 0 | 0 | 24 | 15 |
G:=sub<GL(4,GF(37))| [11,6,0,0,31,17,0,0,0,0,1,0,0,0,0,1],[9,29,0,0,1,28,0,0,0,0,36,0,0,0,0,36],[20,11,0,0,31,17,0,0,0,0,22,11,0,0,3,15],[1,0,0,0,0,1,0,0,0,0,22,24,0,0,3,15] >;
Dic9⋊4D4 in GAP, Magma, Sage, TeX
{\rm Dic}_9\rtimes_4D_4 % in TeX
G:=Group("Dic9:4D4"); // GroupNames label
G:=SmallGroup(288,91);
// by ID
G=gap.SmallGroup(288,91);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,219,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^18=c^4=d^2=1,b^2=a^9,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations