metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36.8D4, D4⋊1Dic9, C18.12D8, C18.6SD16, (D4×C9)⋊1C4, C36.7(C2×C4), (C6×D4).2S3, (C2×D4).2D9, C9⋊3(D4⋊C4), C4⋊Dic9⋊10C2, C2.3(D4⋊D9), (D4×C18).2C2, (C2×C18).34D4, (C2×C4).41D18, (C2×C12).46D6, C4.1(C2×Dic9), C6.19(D4⋊S3), C4.13(C9⋊D4), C12.6(C3⋊D4), C2.3(D4.D9), C6.9(D4.S3), C3.(D4⋊Dic3), (C3×D4).1Dic3, C12.1(C2×Dic3), (C2×C36).24C22, C18.14(C22⋊C4), C22.17(C9⋊D4), C2.4(C18.D4), C6.15(C6.D4), (C2×C9⋊C8)⋊2C2, (C2×C6).72(C3⋊D4), SmallGroup(288,40)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊Dic9
G = < a,b,c,d | a4=b2=c18=1, d2=c9, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Subgroups: 248 in 75 conjugacy classes, 36 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C6 [×3], C6 [×2], C8, C2×C4, C2×C4, D4 [×2], D4, C23, C9, Dic3, C12 [×2], C2×C6, C2×C6 [×4], C4⋊C4, C2×C8, C2×D4, C18 [×3], C18 [×2], C3⋊C8, C2×Dic3, C2×C12, C3×D4 [×2], C3×D4, C22×C6, D4⋊C4, Dic9, C36 [×2], C2×C18, C2×C18 [×4], C2×C3⋊C8, C4⋊Dic3, C6×D4, C9⋊C8, C2×Dic9, C2×C36, D4×C9 [×2], D4×C9, C22×C18, D4⋊Dic3, C2×C9⋊C8, C4⋊Dic9, D4×C18, D4⋊Dic9
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, D8, SD16, D9, C2×Dic3, C3⋊D4 [×2], D4⋊C4, Dic9 [×2], D18, D4⋊S3, D4.S3, C6.D4, C2×Dic9, C9⋊D4 [×2], D4⋊Dic3, D4.D9, D4⋊D9, C18.D4, D4⋊Dic9
►On 144 points
Generators in S144
(1 63 87 116)(2 64 88 117)(3 65 89 118)(4 66 90 119)(5 67 73 120)(6 68 74 121)(7 69 75 122)(8 70 76 123)(9 71 77 124)(10 72 78 125)(11 55 79 126)(12 56 80 109)(13 57 81 110)(14 58 82 111)(15 59 83 112)(16 60 84 113)(17 61 85 114)(18 62 86 115)(19 136 99 37)(20 137 100 38)(21 138 101 39)(22 139 102 40)(23 140 103 41)(24 141 104 42)(25 142 105 43)(26 143 106 44)(27 144 107 45)(28 127 108 46)(29 128 91 47)(30 129 92 48)(31 130 93 49)(32 131 94 50)(33 132 95 51)(34 133 96 52)(35 134 97 53)(36 135 98 54)
(1 125)(2 126)(3 109)(4 110)(5 111)(6 112)(7 113)(8 114)(9 115)(10 116)(11 117)(12 118)(13 119)(14 120)(15 121)(16 122)(17 123)(18 124)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(37 127)(38 128)(39 129)(40 130)(41 131)(42 132)(43 133)(44 134)(45 135)(46 136)(47 137)(48 138)(49 139)(50 140)(51 141)(52 142)(53 143)(54 144)(55 88)(56 89)(57 90)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)(99 108)
(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)
(1 140 10 131)(2 139 11 130)(3 138 12 129)(4 137 13 128)(5 136 14 127)(6 135 15 144)(7 134 16 143)(8 133 17 142)(9 132 18 141)(19 58 28 67)(20 57 29 66)(21 56 30 65)(22 55 31 64)(23 72 32 63)(24 71 33 62)(25 70 34 61)(26 69 35 60)(27 68 36 59)(37 82 46 73)(38 81 47 90)(39 80 48 89)(40 79 49 88)(41 78 50 87)(42 77 51 86)(43 76 52 85)(44 75 53 84)(45 74 54 83)(91 119 100 110)(92 118 101 109)(93 117 102 126)(94 116 103 125)(95 115 104 124)(96 114 105 123)(97 113 106 122)(98 112 107 121)(99 111 108 120)
G:=sub<Sym(144)| (1,63,87,116)(2,64,88,117)(3,65,89,118)(4,66,90,119)(5,67,73,120)(6,68,74,121)(7,69,75,122)(8,70,76,123)(9,71,77,124)(10,72,78,125)(11,55,79,126)(12,56,80,109)(13,57,81,110)(14,58,82,111)(15,59,83,112)(16,60,84,113)(17,61,85,114)(18,62,86,115)(19,136,99,37)(20,137,100,38)(21,138,101,39)(22,139,102,40)(23,140,103,41)(24,141,104,42)(25,142,105,43)(26,143,106,44)(27,144,107,45)(28,127,108,46)(29,128,91,47)(30,129,92,48)(31,130,93,49)(32,131,94,50)(33,132,95,51)(34,133,96,52)(35,134,97,53)(36,135,98,54), (1,125)(2,126)(3,109)(4,110)(5,111)(6,112)(7,113)(8,114)(9,115)(10,116)(11,117)(12,118)(13,119)(14,120)(15,121)(16,122)(17,123)(18,124)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,127)(38,128)(39,129)(40,130)(41,131)(42,132)(43,133)(44,134)(45,135)(46,136)(47,137)(48,138)(49,139)(50,140)(51,141)(52,142)(53,143)(54,144)(55,88)(56,89)(57,90)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107)(99,108), (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), (1,140,10,131)(2,139,11,130)(3,138,12,129)(4,137,13,128)(5,136,14,127)(6,135,15,144)(7,134,16,143)(8,133,17,142)(9,132,18,141)(19,58,28,67)(20,57,29,66)(21,56,30,65)(22,55,31,64)(23,72,32,63)(24,71,33,62)(25,70,34,61)(26,69,35,60)(27,68,36,59)(37,82,46,73)(38,81,47,90)(39,80,48,89)(40,79,49,88)(41,78,50,87)(42,77,51,86)(43,76,52,85)(44,75,53,84)(45,74,54,83)(91,119,100,110)(92,118,101,109)(93,117,102,126)(94,116,103,125)(95,115,104,124)(96,114,105,123)(97,113,106,122)(98,112,107,121)(99,111,108,120)>;
G:=Group( (1,63,87,116)(2,64,88,117)(3,65,89,118)(4,66,90,119)(5,67,73,120)(6,68,74,121)(7,69,75,122)(8,70,76,123)(9,71,77,124)(10,72,78,125)(11,55,79,126)(12,56,80,109)(13,57,81,110)(14,58,82,111)(15,59,83,112)(16,60,84,113)(17,61,85,114)(18,62,86,115)(19,136,99,37)(20,137,100,38)(21,138,101,39)(22,139,102,40)(23,140,103,41)(24,141,104,42)(25,142,105,43)(26,143,106,44)(27,144,107,45)(28,127,108,46)(29,128,91,47)(30,129,92,48)(31,130,93,49)(32,131,94,50)(33,132,95,51)(34,133,96,52)(35,134,97,53)(36,135,98,54), (1,125)(2,126)(3,109)(4,110)(5,111)(6,112)(7,113)(8,114)(9,115)(10,116)(11,117)(12,118)(13,119)(14,120)(15,121)(16,122)(17,123)(18,124)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,127)(38,128)(39,129)(40,130)(41,131)(42,132)(43,133)(44,134)(45,135)(46,136)(47,137)(48,138)(49,139)(50,140)(51,141)(52,142)(53,143)(54,144)(55,88)(56,89)(57,90)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107)(99,108), (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), (1,140,10,131)(2,139,11,130)(3,138,12,129)(4,137,13,128)(5,136,14,127)(6,135,15,144)(7,134,16,143)(8,133,17,142)(9,132,18,141)(19,58,28,67)(20,57,29,66)(21,56,30,65)(22,55,31,64)(23,72,32,63)(24,71,33,62)(25,70,34,61)(26,69,35,60)(27,68,36,59)(37,82,46,73)(38,81,47,90)(39,80,48,89)(40,79,49,88)(41,78,50,87)(42,77,51,86)(43,76,52,85)(44,75,53,84)(45,74,54,83)(91,119,100,110)(92,118,101,109)(93,117,102,126)(94,116,103,125)(95,115,104,124)(96,114,105,123)(97,113,106,122)(98,112,107,121)(99,111,108,120) );
G=PermutationGroup([(1,63,87,116),(2,64,88,117),(3,65,89,118),(4,66,90,119),(5,67,73,120),(6,68,74,121),(7,69,75,122),(8,70,76,123),(9,71,77,124),(10,72,78,125),(11,55,79,126),(12,56,80,109),(13,57,81,110),(14,58,82,111),(15,59,83,112),(16,60,84,113),(17,61,85,114),(18,62,86,115),(19,136,99,37),(20,137,100,38),(21,138,101,39),(22,139,102,40),(23,140,103,41),(24,141,104,42),(25,142,105,43),(26,143,106,44),(27,144,107,45),(28,127,108,46),(29,128,91,47),(30,129,92,48),(31,130,93,49),(32,131,94,50),(33,132,95,51),(34,133,96,52),(35,134,97,53),(36,135,98,54)], [(1,125),(2,126),(3,109),(4,110),(5,111),(6,112),(7,113),(8,114),(9,115),(10,116),(11,117),(12,118),(13,119),(14,120),(15,121),(16,122),(17,123),(18,124),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(37,127),(38,128),(39,129),(40,130),(41,131),(42,132),(43,133),(44,134),(45,135),(46,136),(47,137),(48,138),(49,139),(50,140),(51,141),(52,142),(53,143),(54,144),(55,88),(56,89),(57,90),(58,73),(59,74),(60,75),(61,76),(62,77),(63,78),(64,79),(65,80),(66,81),(67,82),(68,83),(69,84),(70,85),(71,86),(72,87),(91,100),(92,101),(93,102),(94,103),(95,104),(96,105),(97,106),(98,107),(99,108)], [(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)], [(1,140,10,131),(2,139,11,130),(3,138,12,129),(4,137,13,128),(5,136,14,127),(6,135,15,144),(7,134,16,143),(8,133,17,142),(9,132,18,141),(19,58,28,67),(20,57,29,66),(21,56,30,65),(22,55,31,64),(23,72,32,63),(24,71,33,62),(25,70,34,61),(26,69,35,60),(27,68,36,59),(37,82,46,73),(38,81,47,90),(39,80,48,89),(40,79,49,88),(41,78,50,87),(42,77,51,86),(43,76,52,85),(44,75,53,84),(45,74,54,83),(91,119,100,110),(92,118,101,109),(93,117,102,126),(94,116,103,125),(95,115,104,124),(96,114,105,123),(97,113,106,122),(98,112,107,121),(99,111,108,120)])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 18A | ··· | 18I | 18J | ··· | 18U | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | Dic3 | D8 | SD16 | D9 | C3⋊D4 | C3⋊D4 | D18 | Dic9 | C9⋊D4 | C9⋊D4 | D4⋊S3 | D4.S3 | D4.D9 | D4⋊D9 |
| kernel | D4⋊Dic9 | C2×C9⋊C8 | C4⋊Dic9 | D4×C18 | D4×C9 | C6×D4 | C36 | C2×C18 | C2×C12 | C3×D4 | C18 | C18 | C2×D4 | C12 | C2×C6 | C2×C4 | D4 | C4 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 1 | 1 | 3 | 3 |
Matrix representation of D4⋊Dic9 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 29 |
| 0 | 0 | 10 | 72 |
| 1 | 0 | 0 | 0 |
| 16 | 72 | 0 | 0 |
| 0 | 0 | 1 | 29 |
| 0 | 0 | 0 | 72 |
| 36 | 0 | 0 | 0 |
| 12 | 71 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 18 | 0 | 0 |
| 20 | 71 | 0 | 0 |
| 0 | 0 | 0 | 26 |
| 0 | 0 | 59 | 0 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,10,0,0,29,72],[1,16,0,0,0,72,0,0,0,0,1,0,0,0,29,72],[36,12,0,0,0,71,0,0,0,0,1,0,0,0,0,1],[2,20,0,0,18,71,0,0,0,0,0,59,0,0,26,0] >;
D4⋊Dic9 in GAP, Magma, Sage, TeX
D_4\rtimes {\rm Dic}_9 % in TeX
G:=Group("D4:Dic9"); // GroupNames label
G:=SmallGroup(288,40);
// by ID
G=gap.SmallGroup(288,40);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,675,346,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^18=1,d^2=c^9,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations