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 [×6], C2 [×4], C3, C4 [×4], C22, C22 [×10], C22 [×12], C6, C6 [×6], C6 [×4], C2×C4 [×8], C23, C23 [×6], C23 [×4], C9, Dic3 [×4], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×4], C22×C4 [×2], C24, C18, C18 [×6], C18 [×4], C2×Dic3 [×8], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4, Dic9 [×4], C2×C18, C2×C18 [×10], C2×C18 [×12], C6.D4 [×4], C22×Dic3 [×2], C23×C6, C2×Dic9 [×4], C2×Dic9 [×4], C22×C18, C22×C18 [×6], C22×C18 [×4], C2×C6.D4, C18.D4 [×4], C22×Dic9 [×2], C23×C18, C2×C18.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], D9, C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, Dic9 [×4], D18 [×3], C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C2×Dic9 [×6], C9⋊D4 [×4], C22×D9, C2×C6.D4, C18.D4 [×4], C22×Dic9, C2×C9⋊D4 [×2], C2×C18.D4
►On 144 points
Generators in S144
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(37 69)(38 70)(39 71)(40 72)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(73 144)(74 127)(75 128)(76 129)(77 130)(78 131)(79 132)(80 133)(81 134)(82 135)(83 136)(84 137)(85 138)(86 139)(87 140)(88 141)(89 142)(90 143)(91 122)(92 123)(93 124)(94 125)(95 126)(96 109)(97 110)(98 111)(99 112)(100 113)(101 114)(102 115)(103 116)(104 117)(105 118)(106 119)(107 120)(108 121)
(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 48 113 82)(2 47 114 81)(3 46 115 80)(4 45 116 79)(5 44 117 78)(6 43 118 77)(7 42 119 76)(8 41 120 75)(9 40 121 74)(10 39 122 73)(11 38 123 90)(12 37 124 89)(13 54 125 88)(14 53 126 87)(15 52 109 86)(16 51 110 85)(17 50 111 84)(18 49 112 83)(19 70 92 143)(20 69 93 142)(21 68 94 141)(22 67 95 140)(23 66 96 139)(24 65 97 138)(25 64 98 137)(26 63 99 136)(27 62 100 135)(28 61 101 134)(29 60 102 133)(30 59 103 132)(31 58 104 131)(32 57 105 130)(33 56 106 129)(34 55 107 128)(35 72 108 127)(36 71 91 144)
(1 144 10 135)(2 143 11 134)(3 142 12 133)(4 141 13 132)(5 140 14 131)(6 139 15 130)(7 138 16 129)(8 137 17 128)(9 136 18 127)(19 81 28 90)(20 80 29 89)(21 79 30 88)(22 78 31 87)(23 77 32 86)(24 76 33 85)(25 75 34 84)(26 74 35 83)(27 73 36 82)(37 93 46 102)(38 92 47 101)(39 91 48 100)(40 108 49 99)(41 107 50 98)(42 106 51 97)(43 105 52 96)(44 104 53 95)(45 103 54 94)(55 120 64 111)(56 119 65 110)(57 118 66 109)(58 117 67 126)(59 116 68 125)(60 115 69 124)(61 114 70 123)(62 113 71 122)(63 112 72 121)
G:=sub<Sym(144)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(37,69)(38,70)(39,71)(40,72)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(73,144)(74,127)(75,128)(76,129)(77,130)(78,131)(79,132)(80,133)(81,134)(82,135)(83,136)(84,137)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143)(91,122)(92,123)(93,124)(94,125)(95,126)(96,109)(97,110)(98,111)(99,112)(100,113)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,121), (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), (1,48,113,82)(2,47,114,81)(3,46,115,80)(4,45,116,79)(5,44,117,78)(6,43,118,77)(7,42,119,76)(8,41,120,75)(9,40,121,74)(10,39,122,73)(11,38,123,90)(12,37,124,89)(13,54,125,88)(14,53,126,87)(15,52,109,86)(16,51,110,85)(17,50,111,84)(18,49,112,83)(19,70,92,143)(20,69,93,142)(21,68,94,141)(22,67,95,140)(23,66,96,139)(24,65,97,138)(25,64,98,137)(26,63,99,136)(27,62,100,135)(28,61,101,134)(29,60,102,133)(30,59,103,132)(31,58,104,131)(32,57,105,130)(33,56,106,129)(34,55,107,128)(35,72,108,127)(36,71,91,144), (1,144,10,135)(2,143,11,134)(3,142,12,133)(4,141,13,132)(5,140,14,131)(6,139,15,130)(7,138,16,129)(8,137,17,128)(9,136,18,127)(19,81,28,90)(20,80,29,89)(21,79,30,88)(22,78,31,87)(23,77,32,86)(24,76,33,85)(25,75,34,84)(26,74,35,83)(27,73,36,82)(37,93,46,102)(38,92,47,101)(39,91,48,100)(40,108,49,99)(41,107,50,98)(42,106,51,97)(43,105,52,96)(44,104,53,95)(45,103,54,94)(55,120,64,111)(56,119,65,110)(57,118,66,109)(58,117,67,126)(59,116,68,125)(60,115,69,124)(61,114,70,123)(62,113,71,122)(63,112,72,121)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(37,69)(38,70)(39,71)(40,72)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(73,144)(74,127)(75,128)(76,129)(77,130)(78,131)(79,132)(80,133)(81,134)(82,135)(83,136)(84,137)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143)(91,122)(92,123)(93,124)(94,125)(95,126)(96,109)(97,110)(98,111)(99,112)(100,113)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,121), (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), (1,48,113,82)(2,47,114,81)(3,46,115,80)(4,45,116,79)(5,44,117,78)(6,43,118,77)(7,42,119,76)(8,41,120,75)(9,40,121,74)(10,39,122,73)(11,38,123,90)(12,37,124,89)(13,54,125,88)(14,53,126,87)(15,52,109,86)(16,51,110,85)(17,50,111,84)(18,49,112,83)(19,70,92,143)(20,69,93,142)(21,68,94,141)(22,67,95,140)(23,66,96,139)(24,65,97,138)(25,64,98,137)(26,63,99,136)(27,62,100,135)(28,61,101,134)(29,60,102,133)(30,59,103,132)(31,58,104,131)(32,57,105,130)(33,56,106,129)(34,55,107,128)(35,72,108,127)(36,71,91,144), (1,144,10,135)(2,143,11,134)(3,142,12,133)(4,141,13,132)(5,140,14,131)(6,139,15,130)(7,138,16,129)(8,137,17,128)(9,136,18,127)(19,81,28,90)(20,80,29,89)(21,79,30,88)(22,78,31,87)(23,77,32,86)(24,76,33,85)(25,75,34,84)(26,74,35,83)(27,73,36,82)(37,93,46,102)(38,92,47,101)(39,91,48,100)(40,108,49,99)(41,107,50,98)(42,106,51,97)(43,105,52,96)(44,104,53,95)(45,103,54,94)(55,120,64,111)(56,119,65,110)(57,118,66,109)(58,117,67,126)(59,116,68,125)(60,115,69,124)(61,114,70,123)(62,113,71,122)(63,112,72,121) );
G=PermutationGroup([(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26),(37,69),(38,70),(39,71),(40,72),(41,55),(42,56),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(49,63),(50,64),(51,65),(52,66),(53,67),(54,68),(73,144),(74,127),(75,128),(76,129),(77,130),(78,131),(79,132),(80,133),(81,134),(82,135),(83,136),(84,137),(85,138),(86,139),(87,140),(88,141),(89,142),(90,143),(91,122),(92,123),(93,124),(94,125),(95,126),(96,109),(97,110),(98,111),(99,112),(100,113),(101,114),(102,115),(103,116),(104,117),(105,118),(106,119),(107,120),(108,121)], [(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)], [(1,48,113,82),(2,47,114,81),(3,46,115,80),(4,45,116,79),(5,44,117,78),(6,43,118,77),(7,42,119,76),(8,41,120,75),(9,40,121,74),(10,39,122,73),(11,38,123,90),(12,37,124,89),(13,54,125,88),(14,53,126,87),(15,52,109,86),(16,51,110,85),(17,50,111,84),(18,49,112,83),(19,70,92,143),(20,69,93,142),(21,68,94,141),(22,67,95,140),(23,66,96,139),(24,65,97,138),(25,64,98,137),(26,63,99,136),(27,62,100,135),(28,61,101,134),(29,60,102,133),(30,59,103,132),(31,58,104,131),(32,57,105,130),(33,56,106,129),(34,55,107,128),(35,72,108,127),(36,71,91,144)], [(1,144,10,135),(2,143,11,134),(3,142,12,133),(4,141,13,132),(5,140,14,131),(6,139,15,130),(7,138,16,129),(8,137,17,128),(9,136,18,127),(19,81,28,90),(20,80,29,89),(21,79,30,88),(22,78,31,87),(23,77,32,86),(24,76,33,85),(25,75,34,84),(26,74,35,83),(27,73,36,82),(37,93,46,102),(38,92,47,101),(39,91,48,100),(40,108,49,99),(41,107,50,98),(42,106,51,97),(43,105,52,96),(44,104,53,95),(45,103,54,94),(55,120,64,111),(56,119,65,110),(57,118,66,109),(58,117,67,126),(59,116,68,125),(60,115,69,124),(61,114,70,123),(62,113,71,122),(63,112,72,121)])
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