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, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C9, Dic3, C12, C2×C6, C2×Q8, C18, C18, Dic6, C2×Dic3, C2×C12, Dic9, C36, C2×C18, C2×Dic6, Dic18, C2×Dic9, C2×C36, C2×Dic18
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, D9, Dic6, C22×S3, D18, C2×Dic6, Dic18, C22×D9, C2×Dic18
►Regular action on 144 points
Generators in S144
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 121)(82 122)(83 123)(84 124)(85 125)(86 126)(87 127)(88 128)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 109)(106 110)(107 111)(108 112)
(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 90 19 108)(2 89 20 107)(3 88 21 106)(4 87 22 105)(5 86 23 104)(6 85 24 103)(7 84 25 102)(8 83 26 101)(9 82 27 100)(10 81 28 99)(11 80 29 98)(12 79 30 97)(13 78 31 96)(14 77 32 95)(15 76 33 94)(16 75 34 93)(17 74 35 92)(18 73 36 91)(37 112 55 130)(38 111 56 129)(39 110 57 128)(40 109 58 127)(41 144 59 126)(42 143 60 125)(43 142 61 124)(44 141 62 123)(45 140 63 122)(46 139 64 121)(47 138 65 120)(48 137 66 119)(49 136 67 118)(50 135 68 117)(51 134 69 116)(52 133 70 115)(53 132 71 114)(54 131 72 113)
G:=sub<Sym(144)| (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,109)(106,110)(107,111)(108,112), (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), (1,90,19,108)(2,89,20,107)(3,88,21,106)(4,87,22,105)(5,86,23,104)(6,85,24,103)(7,84,25,102)(8,83,26,101)(9,82,27,100)(10,81,28,99)(11,80,29,98)(12,79,30,97)(13,78,31,96)(14,77,32,95)(15,76,33,94)(16,75,34,93)(17,74,35,92)(18,73,36,91)(37,112,55,130)(38,111,56,129)(39,110,57,128)(40,109,58,127)(41,144,59,126)(42,143,60,125)(43,142,61,124)(44,141,62,123)(45,140,63,122)(46,139,64,121)(47,138,65,120)(48,137,66,119)(49,136,67,118)(50,135,68,117)(51,134,69,116)(52,133,70,115)(53,132,71,114)(54,131,72,113)>;
G:=Group( (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,109)(106,110)(107,111)(108,112), (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), (1,90,19,108)(2,89,20,107)(3,88,21,106)(4,87,22,105)(5,86,23,104)(6,85,24,103)(7,84,25,102)(8,83,26,101)(9,82,27,100)(10,81,28,99)(11,80,29,98)(12,79,30,97)(13,78,31,96)(14,77,32,95)(15,76,33,94)(16,75,34,93)(17,74,35,92)(18,73,36,91)(37,112,55,130)(38,111,56,129)(39,110,57,128)(40,109,58,127)(41,144,59,126)(42,143,60,125)(43,142,61,124)(44,141,62,123)(45,140,63,122)(46,139,64,121)(47,138,65,120)(48,137,66,119)(49,136,67,118)(50,135,68,117)(51,134,69,116)(52,133,70,115)(53,132,71,114)(54,131,72,113) );
G=PermutationGroup([[(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54),(73,113),(74,114),(75,115),(76,116),(77,117),(78,118),(79,119),(80,120),(81,121),(82,122),(83,123),(84,124),(85,125),(86,126),(87,127),(88,128),(89,129),(90,130),(91,131),(92,132),(93,133),(94,134),(95,135),(96,136),(97,137),(98,138),(99,139),(100,140),(101,141),(102,142),(103,143),(104,144),(105,109),(106,110),(107,111),(108,112)], [(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)], [(1,90,19,108),(2,89,20,107),(3,88,21,106),(4,87,22,105),(5,86,23,104),(6,85,24,103),(7,84,25,102),(8,83,26,101),(9,82,27,100),(10,81,28,99),(11,80,29,98),(12,79,30,97),(13,78,31,96),(14,77,32,95),(15,76,33,94),(16,75,34,93),(17,74,35,92),(18,73,36,91),(37,112,55,130),(38,111,56,129),(39,110,57,128),(40,109,58,127),(41,144,59,126),(42,143,60,125),(43,142,61,124),(44,141,62,123),(45,140,63,122),(46,139,64,121),(47,138,65,120),(48,137,66,119),(49,136,67,118),(50,135,68,117),(51,134,69,116),(52,133,70,115),(53,132,71,114),(54,131,72,113)]])
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