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, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, C18, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4×D4, Dic9, Dic9, C36, D18, D18, C2×C18, C2×C18, C2×C18, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C4×D9, C2×Dic9, C2×Dic9, C9⋊D4, C2×C36, 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, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, D9, C4×S3, C22×S3, C4×D4, D18, S3×C2×C4, S3×D4, D4⋊2S3, C4×D9, 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)
(1 143 10 134)(2 142 11 133)(3 141 12 132)(4 140 13 131)(5 139 14 130)(6 138 15 129)(7 137 16 128)(8 136 17 127)(9 135 18 144)(19 117 28 126)(20 116 29 125)(21 115 30 124)(22 114 31 123)(23 113 32 122)(24 112 33 121)(25 111 34 120)(26 110 35 119)(27 109 36 118)(37 77 46 86)(38 76 47 85)(39 75 48 84)(40 74 49 83)(41 73 50 82)(42 90 51 81)(43 89 52 80)(44 88 53 79)(45 87 54 78)(55 105 64 96)(56 104 65 95)(57 103 66 94)(58 102 67 93)(59 101 68 92)(60 100 69 91)(61 99 70 108)(62 98 71 107)(63 97 72 106)
(1 43 34 107)(2 42 35 106)(3 41 36 105)(4 40 19 104)(5 39 20 103)(6 38 21 102)(7 37 22 101)(8 54 23 100)(9 53 24 99)(10 52 25 98)(11 51 26 97)(12 50 27 96)(13 49 28 95)(14 48 29 94)(15 47 30 93)(16 46 31 92)(17 45 32 91)(18 44 33 108)(55 132 82 109)(56 131 83 126)(57 130 84 125)(58 129 85 124)(59 128 86 123)(60 127 87 122)(61 144 88 121)(62 143 89 120)(63 142 90 119)(64 141 73 118)(65 140 74 117)(66 139 75 116)(67 138 76 115)(68 137 77 114)(69 136 78 113)(70 135 79 112)(71 134 80 111)(72 133 81 110)
(1 107)(2 108)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)(73 122)(74 123)(75 124)(76 125)(77 126)(78 109)(79 110)(80 111)(81 112)(82 113)(83 114)(84 115)(85 116)(86 117)(87 118)(88 119)(89 120)(90 121)
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), (1,143,10,134)(2,142,11,133)(3,141,12,132)(4,140,13,131)(5,139,14,130)(6,138,15,129)(7,137,16,128)(8,136,17,127)(9,135,18,144)(19,117,28,126)(20,116,29,125)(21,115,30,124)(22,114,31,123)(23,113,32,122)(24,112,33,121)(25,111,34,120)(26,110,35,119)(27,109,36,118)(37,77,46,86)(38,76,47,85)(39,75,48,84)(40,74,49,83)(41,73,50,82)(42,90,51,81)(43,89,52,80)(44,88,53,79)(45,87,54,78)(55,105,64,96)(56,104,65,95)(57,103,66,94)(58,102,67,93)(59,101,68,92)(60,100,69,91)(61,99,70,108)(62,98,71,107)(63,97,72,106), (1,43,34,107)(2,42,35,106)(3,41,36,105)(4,40,19,104)(5,39,20,103)(6,38,21,102)(7,37,22,101)(8,54,23,100)(9,53,24,99)(10,52,25,98)(11,51,26,97)(12,50,27,96)(13,49,28,95)(14,48,29,94)(15,47,30,93)(16,46,31,92)(17,45,32,91)(18,44,33,108)(55,132,82,109)(56,131,83,126)(57,130,84,125)(58,129,85,124)(59,128,86,123)(60,127,87,122)(61,144,88,121)(62,143,89,120)(63,142,90,119)(64,141,73,118)(65,140,74,117)(66,139,75,116)(67,138,76,115)(68,137,77,114)(69,136,78,113)(70,135,79,112)(71,134,80,111)(72,133,81,110), (1,107)(2,108)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,122)(74,123)(75,124)(76,125)(77,126)(78,109)(79,110)(80,111)(81,112)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)>;
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), (1,143,10,134)(2,142,11,133)(3,141,12,132)(4,140,13,131)(5,139,14,130)(6,138,15,129)(7,137,16,128)(8,136,17,127)(9,135,18,144)(19,117,28,126)(20,116,29,125)(21,115,30,124)(22,114,31,123)(23,113,32,122)(24,112,33,121)(25,111,34,120)(26,110,35,119)(27,109,36,118)(37,77,46,86)(38,76,47,85)(39,75,48,84)(40,74,49,83)(41,73,50,82)(42,90,51,81)(43,89,52,80)(44,88,53,79)(45,87,54,78)(55,105,64,96)(56,104,65,95)(57,103,66,94)(58,102,67,93)(59,101,68,92)(60,100,69,91)(61,99,70,108)(62,98,71,107)(63,97,72,106), (1,43,34,107)(2,42,35,106)(3,41,36,105)(4,40,19,104)(5,39,20,103)(6,38,21,102)(7,37,22,101)(8,54,23,100)(9,53,24,99)(10,52,25,98)(11,51,26,97)(12,50,27,96)(13,49,28,95)(14,48,29,94)(15,47,30,93)(16,46,31,92)(17,45,32,91)(18,44,33,108)(55,132,82,109)(56,131,83,126)(57,130,84,125)(58,129,85,124)(59,128,86,123)(60,127,87,122)(61,144,88,121)(62,143,89,120)(63,142,90,119)(64,141,73,118)(65,140,74,117)(66,139,75,116)(67,138,76,115)(68,137,77,114)(69,136,78,113)(70,135,79,112)(71,134,80,111)(72,133,81,110), (1,107)(2,108)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,122)(74,123)(75,124)(76,125)(77,126)(78,109)(79,110)(80,111)(81,112)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121) );
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)], [(1,143,10,134),(2,142,11,133),(3,141,12,132),(4,140,13,131),(5,139,14,130),(6,138,15,129),(7,137,16,128),(8,136,17,127),(9,135,18,144),(19,117,28,126),(20,116,29,125),(21,115,30,124),(22,114,31,123),(23,113,32,122),(24,112,33,121),(25,111,34,120),(26,110,35,119),(27,109,36,118),(37,77,46,86),(38,76,47,85),(39,75,48,84),(40,74,49,83),(41,73,50,82),(42,90,51,81),(43,89,52,80),(44,88,53,79),(45,87,54,78),(55,105,64,96),(56,104,65,95),(57,103,66,94),(58,102,67,93),(59,101,68,92),(60,100,69,91),(61,99,70,108),(62,98,71,107),(63,97,72,106)], [(1,43,34,107),(2,42,35,106),(3,41,36,105),(4,40,19,104),(5,39,20,103),(6,38,21,102),(7,37,22,101),(8,54,23,100),(9,53,24,99),(10,52,25,98),(11,51,26,97),(12,50,27,96),(13,49,28,95),(14,48,29,94),(15,47,30,93),(16,46,31,92),(17,45,32,91),(18,44,33,108),(55,132,82,109),(56,131,83,126),(57,130,84,125),(58,129,85,124),(59,128,86,123),(60,127,87,122),(61,144,88,121),(62,143,89,120),(63,142,90,119),(64,141,73,118),(65,140,74,117),(66,139,75,116),(67,138,76,115),(68,137,77,114),(69,136,78,113),(70,135,79,112),(71,134,80,111),(72,133,81,110)], [(1,107),(2,108),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,97),(10,98),(11,99),(12,100),(13,101),(14,102),(15,103),(16,104),(17,105),(18,106),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,37),(29,38),(30,39),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45),(55,136),(56,137),(57,138),(58,139),(59,140),(60,141),(61,142),(62,143),(63,144),(64,127),(65,128),(66,129),(67,130),(68,131),(69,132),(70,133),(71,134),(72,135),(73,122),(74,123),(75,124),(76,125),(77,126),(78,109),(79,110),(80,111),(81,112),(82,113),(83,114),(84,115),(85,116),(86,117),(87,118),(88,119),(89,120),(90,121)]])
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