direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic18, C18⋊Q8, C4.11D18, C12.43D6, C6.5Dic6, C18.1C23, C22.8D18, C36.11C22, Dic9.1C22, C9⋊1(C2×Q8), (C2×C4).4D9, C3.(C2×Dic6), (C2×C36).3C2, (C2×C12).6S3, (C2×C6).24D6, C2.3(C22×D9), C6.19(C22×S3), (C2×C18).8C22, (C2×Dic9).3C2, SmallGroup(144,37)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic18
G = < a,b,c | a2=b36=1, c2=b18, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 167 in 57 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×4], C22, C6, C6 [×2], C2×C4, C2×C4 [×2], Q8 [×4], C9, Dic3 [×4], C12 [×2], C2×C6, C2×Q8, C18, C18 [×2], Dic6 [×4], C2×Dic3 [×2], C2×C12, Dic9 [×4], C36 [×2], C2×C18, C2×Dic6, Dic18 [×4], C2×Dic9 [×2], C2×C36, C2×Dic18
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D6 [×3], C2×Q8, D9, Dic6 [×2], C22×S3, D18 [×3], C2×Dic6, Dic18 [×2], C22×D9, C2×Dic18
►Regular action on 144 points
Generators in S144
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 72)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(73 114)(74 115)(75 116)(76 117)(77 118)(78 119)(79 120)(80 121)(81 122)(82 123)(83 124)(84 125)(85 126)(86 127)(87 128)(88 129)(89 130)(90 131)(91 132)(92 133)(93 134)(94 135)(95 136)(96 137)(97 138)(98 139)(99 140)(100 141)(101 142)(102 143)(103 144)(104 109)(105 110)(106 111)(107 112)(108 113)
(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 99 19 81)(2 98 20 80)(3 97 21 79)(4 96 22 78)(5 95 23 77)(6 94 24 76)(7 93 25 75)(8 92 26 74)(9 91 27 73)(10 90 28 108)(11 89 29 107)(12 88 30 106)(13 87 31 105)(14 86 32 104)(15 85 33 103)(16 84 34 102)(17 83 35 101)(18 82 36 100)(37 115 55 133)(38 114 56 132)(39 113 57 131)(40 112 58 130)(41 111 59 129)(42 110 60 128)(43 109 61 127)(44 144 62 126)(45 143 63 125)(46 142 64 124)(47 141 65 123)(48 140 66 122)(49 139 67 121)(50 138 68 120)(51 137 69 119)(52 136 70 118)(53 135 71 117)(54 134 72 116)
G:=sub<Sym(144)| (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(73,114)(74,115)(75,116)(76,117)(77,118)(78,119)(79,120)(80,121)(81,122)(82,123)(83,124)(84,125)(85,126)(86,127)(87,128)(88,129)(89,130)(90,131)(91,132)(92,133)(93,134)(94,135)(95,136)(96,137)(97,138)(98,139)(99,140)(100,141)(101,142)(102,143)(103,144)(104,109)(105,110)(106,111)(107,112)(108,113), (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), (1,99,19,81)(2,98,20,80)(3,97,21,79)(4,96,22,78)(5,95,23,77)(6,94,24,76)(7,93,25,75)(8,92,26,74)(9,91,27,73)(10,90,28,108)(11,89,29,107)(12,88,30,106)(13,87,31,105)(14,86,32,104)(15,85,33,103)(16,84,34,102)(17,83,35,101)(18,82,36,100)(37,115,55,133)(38,114,56,132)(39,113,57,131)(40,112,58,130)(41,111,59,129)(42,110,60,128)(43,109,61,127)(44,144,62,126)(45,143,63,125)(46,142,64,124)(47,141,65,123)(48,140,66,122)(49,139,67,121)(50,138,68,120)(51,137,69,119)(52,136,70,118)(53,135,71,117)(54,134,72,116)>;
G:=Group( (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(73,114)(74,115)(75,116)(76,117)(77,118)(78,119)(79,120)(80,121)(81,122)(82,123)(83,124)(84,125)(85,126)(86,127)(87,128)(88,129)(89,130)(90,131)(91,132)(92,133)(93,134)(94,135)(95,136)(96,137)(97,138)(98,139)(99,140)(100,141)(101,142)(102,143)(103,144)(104,109)(105,110)(106,111)(107,112)(108,113), (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), (1,99,19,81)(2,98,20,80)(3,97,21,79)(4,96,22,78)(5,95,23,77)(6,94,24,76)(7,93,25,75)(8,92,26,74)(9,91,27,73)(10,90,28,108)(11,89,29,107)(12,88,30,106)(13,87,31,105)(14,86,32,104)(15,85,33,103)(16,84,34,102)(17,83,35,101)(18,82,36,100)(37,115,55,133)(38,114,56,132)(39,113,57,131)(40,112,58,130)(41,111,59,129)(42,110,60,128)(43,109,61,127)(44,144,62,126)(45,143,63,125)(46,142,64,124)(47,141,65,123)(48,140,66,122)(49,139,67,121)(50,138,68,120)(51,137,69,119)(52,136,70,118)(53,135,71,117)(54,134,72,116) );
G=PermutationGroup([(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,68),(22,69),(23,70),(24,71),(25,72),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44),(34,45),(35,46),(36,47),(73,114),(74,115),(75,116),(76,117),(77,118),(78,119),(79,120),(80,121),(81,122),(82,123),(83,124),(84,125),(85,126),(86,127),(87,128),(88,129),(89,130),(90,131),(91,132),(92,133),(93,134),(94,135),(95,136),(96,137),(97,138),(98,139),(99,140),(100,141),(101,142),(102,143),(103,144),(104,109),(105,110),(106,111),(107,112),(108,113)], [(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)], [(1,99,19,81),(2,98,20,80),(3,97,21,79),(4,96,22,78),(5,95,23,77),(6,94,24,76),(7,93,25,75),(8,92,26,74),(9,91,27,73),(10,90,28,108),(11,89,29,107),(12,88,30,106),(13,87,31,105),(14,86,32,104),(15,85,33,103),(16,84,34,102),(17,83,35,101),(18,82,36,100),(37,115,55,133),(38,114,56,132),(39,113,57,131),(40,112,58,130),(41,111,59,129),(42,110,60,128),(43,109,61,127),(44,144,62,126),(45,143,63,125),(46,142,64,124),(47,141,65,123),(48,140,66,122),(49,139,67,121),(50,138,68,120),(51,137,69,119),(52,136,70,118),(53,135,71,117),(54,134,72,116)])
C2×Dic18 is a maximal subgroup of
C18.Q16 C36.45D4 C4.D36 C36⋊2Q8 C42⋊7D9 C22⋊2Dic18 Dic9.D4 Dic9⋊3Q8 C36⋊Q8 D18⋊Q8 D18⋊2Q8 C8.D18 C36.49D4 C36.17D4 Dic9⋊Q8 D4.D18 C2×Q8×D9 D4.10D18
C2×Dic18 is a maximal quotient of
C36⋊2Q8 C36.6Q8 C22⋊2Dic18 C36⋊Q8 C36.3Q8 C36.49D4
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D9 | Dic6 | D18 | D18 | Dic18 |
| kernel | C2×Dic18 | Dic18 | C2×Dic9 | C2×C36 | C2×C12 | C18 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 1 | 2 | 2 | 1 | 3 | 4 | 6 | 3 | 12 |
Matrix representation of C2×Dic18 ►in GL3(𝔽37) generated by
| 36 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 4 | 29 |
| 0 | 8 | 12 |
| 1 | 0 | 0 |
| 0 | 8 | 36 |
| 0 | 28 | 29 |
G:=sub<GL(3,GF(37))| [36,0,0,0,1,0,0,0,1],[1,0,0,0,4,8,0,29,12],[1,0,0,0,8,28,0,36,29] >;
C2×Dic18 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{18} % in TeX
G:=Group("C2xDic18"); // GroupNames label
G:=SmallGroup(144,37);
// by ID
G=gap.SmallGroup(144,37);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,218,50,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c|a^2=b^36=1,c^2=b^18,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations