direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C18.D4, C24.2D9, C23⋊3Dic9, C23.30D18, (C22×C18)⋊4C4, (C2×C18).44D4, C18.62(C2×D4), (C23×C6).8S3, C18⋊2(C22⋊C4), (C23×C18).2C2, C22⋊3(C2×Dic9), C18.28(C22×C4), (C2×C18).60C23, (C2×Dic9)⋊7C22, (C22×Dic9)⋊7C2, (C22×C6).143D6, C2.9(C22×Dic9), C22.25(C9⋊D4), C6.29(C22×Dic3), (C22×C6).14Dic3, C22.27(C22×D9), C6.20(C6.D4), (C22×C18).41C22, C9⋊3(C2×C22⋊C4), (C2×C18)⋊8(C2×C4), C2.4(C2×C9⋊D4), C3.(C2×C6.D4), C6.110(C2×C3⋊D4), (C2×C6).83(C3⋊D4), (C2×C6).41(C2×Dic3), (C2×C6).217(C22×S3), SmallGroup(288,162)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C18.D4
G = < a,b,c,d | a2=b18=c4=1, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b9c-1 >
Subgroups: 552 in 198 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C23, C23, C9, Dic3, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C18, C18, C18, C2×Dic3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, Dic9, C2×C18, C2×C18, C2×C18, C6.D4, C22×Dic3, C23×C6, C2×Dic9, C2×Dic9, C22×C18, C22×C18, C22×C18, C2×C6.D4, C18.D4, C22×Dic9, C23×C18, C2×C18.D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, D9, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, Dic9, D18, C6.D4, C22×Dic3, C2×C3⋊D4, C2×Dic9, C9⋊D4, C22×D9, C2×C6.D4, C18.D4, C22×Dic9, C2×C9⋊D4, C2×C18.D4
►On 144 points
Generators in S144
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 55)(34 56)(35 57)(36 58)(73 131)(74 132)(75 133)(76 134)(77 135)(78 136)(79 137)(80 138)(81 139)(82 140)(83 141)(84 142)(85 143)(86 144)(87 127)(88 128)(89 129)(90 130)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 109)(101 110)(102 111)(103 112)(104 113)(105 114)(106 115)(107 116)(108 117)
(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 82 22 109)(2 81 23 126)(3 80 24 125)(4 79 25 124)(5 78 26 123)(6 77 27 122)(7 76 28 121)(8 75 29 120)(9 74 30 119)(10 73 31 118)(11 90 32 117)(12 89 33 116)(13 88 34 115)(14 87 35 114)(15 86 36 113)(16 85 19 112)(17 84 20 111)(18 83 21 110)(37 131 71 91)(38 130 72 108)(39 129 55 107)(40 128 56 106)(41 127 57 105)(42 144 58 104)(43 143 59 103)(44 142 60 102)(45 141 61 101)(46 140 62 100)(47 139 63 99)(48 138 64 98)(49 137 65 97)(50 136 66 96)(51 135 67 95)(52 134 68 94)(53 133 69 93)(54 132 70 92)
(1 91 10 100)(2 108 11 99)(3 107 12 98)(4 106 13 97)(5 105 14 96)(6 104 15 95)(7 103 16 94)(8 102 17 93)(9 101 18 92)(19 134 28 143)(20 133 29 142)(21 132 30 141)(22 131 31 140)(23 130 32 139)(24 129 33 138)(25 128 34 137)(26 127 35 136)(27 144 36 135)(37 109 46 118)(38 126 47 117)(39 125 48 116)(40 124 49 115)(41 123 50 114)(42 122 51 113)(43 121 52 112)(44 120 53 111)(45 119 54 110)(55 80 64 89)(56 79 65 88)(57 78 66 87)(58 77 67 86)(59 76 68 85)(60 75 69 84)(61 74 70 83)(62 73 71 82)(63 90 72 81)
G:=sub<Sym(144)| (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,55)(34,56)(35,57)(36,58)(73,131)(74,132)(75,133)(76,134)(77,135)(78,136)(79,137)(80,138)(81,139)(82,140)(83,141)(84,142)(85,143)(86,144)(87,127)(88,128)(89,129)(90,130)(91,118)(92,119)(93,120)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,109)(101,110)(102,111)(103,112)(104,113)(105,114)(106,115)(107,116)(108,117), (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), (1,82,22,109)(2,81,23,126)(3,80,24,125)(4,79,25,124)(5,78,26,123)(6,77,27,122)(7,76,28,121)(8,75,29,120)(9,74,30,119)(10,73,31,118)(11,90,32,117)(12,89,33,116)(13,88,34,115)(14,87,35,114)(15,86,36,113)(16,85,19,112)(17,84,20,111)(18,83,21,110)(37,131,71,91)(38,130,72,108)(39,129,55,107)(40,128,56,106)(41,127,57,105)(42,144,58,104)(43,143,59,103)(44,142,60,102)(45,141,61,101)(46,140,62,100)(47,139,63,99)(48,138,64,98)(49,137,65,97)(50,136,66,96)(51,135,67,95)(52,134,68,94)(53,133,69,93)(54,132,70,92), (1,91,10,100)(2,108,11,99)(3,107,12,98)(4,106,13,97)(5,105,14,96)(6,104,15,95)(7,103,16,94)(8,102,17,93)(9,101,18,92)(19,134,28,143)(20,133,29,142)(21,132,30,141)(22,131,31,140)(23,130,32,139)(24,129,33,138)(25,128,34,137)(26,127,35,136)(27,144,36,135)(37,109,46,118)(38,126,47,117)(39,125,48,116)(40,124,49,115)(41,123,50,114)(42,122,51,113)(43,121,52,112)(44,120,53,111)(45,119,54,110)(55,80,64,89)(56,79,65,88)(57,78,66,87)(58,77,67,86)(59,76,68,85)(60,75,69,84)(61,74,70,83)(62,73,71,82)(63,90,72,81)>;
G:=Group( (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,55)(34,56)(35,57)(36,58)(73,131)(74,132)(75,133)(76,134)(77,135)(78,136)(79,137)(80,138)(81,139)(82,140)(83,141)(84,142)(85,143)(86,144)(87,127)(88,128)(89,129)(90,130)(91,118)(92,119)(93,120)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,109)(101,110)(102,111)(103,112)(104,113)(105,114)(106,115)(107,116)(108,117), (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), (1,82,22,109)(2,81,23,126)(3,80,24,125)(4,79,25,124)(5,78,26,123)(6,77,27,122)(7,76,28,121)(8,75,29,120)(9,74,30,119)(10,73,31,118)(11,90,32,117)(12,89,33,116)(13,88,34,115)(14,87,35,114)(15,86,36,113)(16,85,19,112)(17,84,20,111)(18,83,21,110)(37,131,71,91)(38,130,72,108)(39,129,55,107)(40,128,56,106)(41,127,57,105)(42,144,58,104)(43,143,59,103)(44,142,60,102)(45,141,61,101)(46,140,62,100)(47,139,63,99)(48,138,64,98)(49,137,65,97)(50,136,66,96)(51,135,67,95)(52,134,68,94)(53,133,69,93)(54,132,70,92), (1,91,10,100)(2,108,11,99)(3,107,12,98)(4,106,13,97)(5,105,14,96)(6,104,15,95)(7,103,16,94)(8,102,17,93)(9,101,18,92)(19,134,28,143)(20,133,29,142)(21,132,30,141)(22,131,31,140)(23,130,32,139)(24,129,33,138)(25,128,34,137)(26,127,35,136)(27,144,36,135)(37,109,46,118)(38,126,47,117)(39,125,48,116)(40,124,49,115)(41,123,50,114)(42,122,51,113)(43,121,52,112)(44,120,53,111)(45,119,54,110)(55,80,64,89)(56,79,65,88)(57,78,66,87)(58,77,67,86)(59,76,68,85)(60,75,69,84)(61,74,70,83)(62,73,71,82)(63,90,72,81) );
G=PermutationGroup([[(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,55),(34,56),(35,57),(36,58),(73,131),(74,132),(75,133),(76,134),(77,135),(78,136),(79,137),(80,138),(81,139),(82,140),(83,141),(84,142),(85,143),(86,144),(87,127),(88,128),(89,129),(90,130),(91,118),(92,119),(93,120),(94,121),(95,122),(96,123),(97,124),(98,125),(99,126),(100,109),(101,110),(102,111),(103,112),(104,113),(105,114),(106,115),(107,116),(108,117)], [(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)], [(1,82,22,109),(2,81,23,126),(3,80,24,125),(4,79,25,124),(5,78,26,123),(6,77,27,122),(7,76,28,121),(8,75,29,120),(9,74,30,119),(10,73,31,118),(11,90,32,117),(12,89,33,116),(13,88,34,115),(14,87,35,114),(15,86,36,113),(16,85,19,112),(17,84,20,111),(18,83,21,110),(37,131,71,91),(38,130,72,108),(39,129,55,107),(40,128,56,106),(41,127,57,105),(42,144,58,104),(43,143,59,103),(44,142,60,102),(45,141,61,101),(46,140,62,100),(47,139,63,99),(48,138,64,98),(49,137,65,97),(50,136,66,96),(51,135,67,95),(52,134,68,94),(53,133,69,93),(54,132,70,92)], [(1,91,10,100),(2,108,11,99),(3,107,12,98),(4,106,13,97),(5,105,14,96),(6,104,15,95),(7,103,16,94),(8,102,17,93),(9,101,18,92),(19,134,28,143),(20,133,29,142),(21,132,30,141),(22,131,31,140),(23,130,32,139),(24,129,33,138),(25,128,34,137),(26,127,35,136),(27,144,36,135),(37,109,46,118),(38,126,47,117),(39,125,48,116),(40,124,49,115),(41,123,50,114),(42,122,51,113),(43,121,52,112),(44,120,53,111),(45,119,54,110),(55,80,64,89),(56,79,65,88),(57,78,66,87),(58,77,67,86),(59,76,68,85),(60,75,69,84),(61,74,70,83),(62,73,71,82),(63,90,72,81)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4H | 6A | ··· | 6O | 9A | 9B | 9C | 18A | ··· | 18AS |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | 9 | 9 | 18 | ··· | 18 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 18 | ··· | 18 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Dic3 | D6 | D9 | C3⋊D4 | Dic9 | D18 | C9⋊D4 |
| kernel | C2×C18.D4 | C18.D4 | C22×Dic9 | C23×C18 | C22×C18 | C23×C6 | C2×C18 | C22×C6 | C22×C6 | C24 | C2×C6 | C23 | C23 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 1 | 4 | 4 | 3 | 3 | 8 | 12 | 9 | 24 |
Matrix representation of C2×C18.D4 ►in GL5(𝔽37)
| 36 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 0 | 21 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 36 | 0 |
G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,26,0,0,0,0,0,10,0,0,0,0,0,30,0,0,0,0,0,21],[1,0,0,0,0,0,0,36,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,1,0] >;
C2×C18.D4 in GAP, Magma, Sage, TeX
C_2\times C_{18}.D_4 % in TeX
G:=Group("C2xC18.D4"); // GroupNames label
G:=SmallGroup(288,162);
// by ID
G=gap.SmallGroup(288,162);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,422,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^18=c^4=1,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^9*c^-1>;
// generators/relations