direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4.Dic9, C18⋊2M4(2), C36.41C23, C23.4Dic9, C9⋊C8⋊12C22, (C2×C36).9C4, C9⋊3(C2×M4(2)), C36.38(C2×C4), (C2×C4).84D18, (C2×C4).6Dic9, (C22×C4).6D9, (C2×C12).395D6, (C22×C36).9C2, (C22×C18).7C4, C4.14(C2×Dic9), C4.41(C22×D9), (C22×C12).26S3, (C2×C36).98C22, C18.22(C22×C4), (C2×C12).14Dic3, C12.54(C2×Dic3), C6.7(C4.Dic3), C2.3(C22×Dic9), C12.202(C22×S3), C6.22(C22×Dic3), C22.12(C2×Dic9), (C22×C6).13Dic3, (C2×C9⋊C8)⋊12C2, C3.(C2×C4.Dic3), (C2×C18).33(C2×C4), (C2×C6).37(C2×Dic3), SmallGroup(288,131)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4.Dic9
G = < a,b,c,d | a2=b4=1, c18=b2, d2=c9, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c17 >
Subgroups: 224 in 102 conjugacy classes, 68 normal (28 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C9, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C18, C18, C18, C3⋊C8, C2×C12, C2×C12, C22×C6, C2×M4(2), C36, C36, C2×C18, C2×C18, C2×C18, C2×C3⋊C8, C4.Dic3, C22×C12, C9⋊C8, C2×C36, C2×C36, C22×C18, C2×C4.Dic3, C2×C9⋊C8, C4.Dic9, C22×C36, C2×C4.Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, M4(2), C22×C4, D9, C2×Dic3, C22×S3, C2×M4(2), Dic9, D18, C4.Dic3, C22×Dic3, C2×Dic9, C22×D9, C2×C4.Dic3, C4.Dic9, C22×Dic9, C2×C4.Dic9
►On 144 points
Generators in S144
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(73 112)(74 113)(75 114)(76 115)(77 116)(78 117)(79 118)(80 119)(81 120)(82 121)(83 122)(84 123)(85 124)(86 125)(87 126)(88 127)(89 128)(90 129)(91 130)(92 131)(93 132)(94 133)(95 134)(96 135)(97 136)(98 137)(99 138)(100 139)(101 140)(102 141)(103 142)(104 143)(105 144)(106 109)(107 110)(108 111)
(1 10 19 28)(2 11 20 29)(3 12 21 30)(4 13 22 31)(5 14 23 32)(6 15 24 33)(7 16 25 34)(8 17 26 35)(9 18 27 36)(37 46 55 64)(38 47 56 65)(39 48 57 66)(40 49 58 67)(41 50 59 68)(42 51 60 69)(43 52 61 70)(44 53 62 71)(45 54 63 72)(73 100 91 82)(74 101 92 83)(75 102 93 84)(76 103 94 85)(77 104 95 86)(78 105 96 87)(79 106 97 88)(80 107 98 89)(81 108 99 90)(109 136 127 118)(110 137 128 119)(111 138 129 120)(112 139 130 121)(113 140 131 122)(114 141 132 123)(115 142 133 124)(116 143 134 125)(117 144 135 126)
(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)
(1 129 10 138 19 111 28 120)(2 110 11 119 20 128 29 137)(3 127 12 136 21 109 30 118)(4 144 13 117 22 126 31 135)(5 125 14 134 23 143 32 116)(6 142 15 115 24 124 33 133)(7 123 16 132 25 141 34 114)(8 140 17 113 26 122 35 131)(9 121 18 130 27 139 36 112)(37 85 46 94 55 103 64 76)(38 102 47 75 56 84 65 93)(39 83 48 92 57 101 66 74)(40 100 49 73 58 82 67 91)(41 81 50 90 59 99 68 108)(42 98 51 107 60 80 69 89)(43 79 52 88 61 97 70 106)(44 96 53 105 62 78 71 87)(45 77 54 86 63 95 72 104)
G:=sub<Sym(144)| (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,120)(82,121)(83,122)(84,123)(85,124)(86,125)(87,126)(88,127)(89,128)(90,129)(91,130)(92,131)(93,132)(94,133)(95,134)(96,135)(97,136)(98,137)(99,138)(100,139)(101,140)(102,141)(103,142)(104,143)(105,144)(106,109)(107,110)(108,111), (1,10,19,28)(2,11,20,29)(3,12,21,30)(4,13,22,31)(5,14,23,32)(6,15,24,33)(7,16,25,34)(8,17,26,35)(9,18,27,36)(37,46,55,64)(38,47,56,65)(39,48,57,66)(40,49,58,67)(41,50,59,68)(42,51,60,69)(43,52,61,70)(44,53,62,71)(45,54,63,72)(73,100,91,82)(74,101,92,83)(75,102,93,84)(76,103,94,85)(77,104,95,86)(78,105,96,87)(79,106,97,88)(80,107,98,89)(81,108,99,90)(109,136,127,118)(110,137,128,119)(111,138,129,120)(112,139,130,121)(113,140,131,122)(114,141,132,123)(115,142,133,124)(116,143,134,125)(117,144,135,126), (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), (1,129,10,138,19,111,28,120)(2,110,11,119,20,128,29,137)(3,127,12,136,21,109,30,118)(4,144,13,117,22,126,31,135)(5,125,14,134,23,143,32,116)(6,142,15,115,24,124,33,133)(7,123,16,132,25,141,34,114)(8,140,17,113,26,122,35,131)(9,121,18,130,27,139,36,112)(37,85,46,94,55,103,64,76)(38,102,47,75,56,84,65,93)(39,83,48,92,57,101,66,74)(40,100,49,73,58,82,67,91)(41,81,50,90,59,99,68,108)(42,98,51,107,60,80,69,89)(43,79,52,88,61,97,70,106)(44,96,53,105,62,78,71,87)(45,77,54,86,63,95,72,104)>;
G:=Group( (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,120)(82,121)(83,122)(84,123)(85,124)(86,125)(87,126)(88,127)(89,128)(90,129)(91,130)(92,131)(93,132)(94,133)(95,134)(96,135)(97,136)(98,137)(99,138)(100,139)(101,140)(102,141)(103,142)(104,143)(105,144)(106,109)(107,110)(108,111), (1,10,19,28)(2,11,20,29)(3,12,21,30)(4,13,22,31)(5,14,23,32)(6,15,24,33)(7,16,25,34)(8,17,26,35)(9,18,27,36)(37,46,55,64)(38,47,56,65)(39,48,57,66)(40,49,58,67)(41,50,59,68)(42,51,60,69)(43,52,61,70)(44,53,62,71)(45,54,63,72)(73,100,91,82)(74,101,92,83)(75,102,93,84)(76,103,94,85)(77,104,95,86)(78,105,96,87)(79,106,97,88)(80,107,98,89)(81,108,99,90)(109,136,127,118)(110,137,128,119)(111,138,129,120)(112,139,130,121)(113,140,131,122)(114,141,132,123)(115,142,133,124)(116,143,134,125)(117,144,135,126), (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), (1,129,10,138,19,111,28,120)(2,110,11,119,20,128,29,137)(3,127,12,136,21,109,30,118)(4,144,13,117,22,126,31,135)(5,125,14,134,23,143,32,116)(6,142,15,115,24,124,33,133)(7,123,16,132,25,141,34,114)(8,140,17,113,26,122,35,131)(9,121,18,130,27,139,36,112)(37,85,46,94,55,103,64,76)(38,102,47,75,56,84,65,93)(39,83,48,92,57,101,66,74)(40,100,49,73,58,82,67,91)(41,81,50,90,59,99,68,108)(42,98,51,107,60,80,69,89)(43,79,52,88,61,97,70,106)(44,96,53,105,62,78,71,87)(45,77,54,86,63,95,72,104) );
G=PermutationGroup([[(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,37),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(73,112),(74,113),(75,114),(76,115),(77,116),(78,117),(79,118),(80,119),(81,120),(82,121),(83,122),(84,123),(85,124),(86,125),(87,126),(88,127),(89,128),(90,129),(91,130),(92,131),(93,132),(94,133),(95,134),(96,135),(97,136),(98,137),(99,138),(100,139),(101,140),(102,141),(103,142),(104,143),(105,144),(106,109),(107,110),(108,111)], [(1,10,19,28),(2,11,20,29),(3,12,21,30),(4,13,22,31),(5,14,23,32),(6,15,24,33),(7,16,25,34),(8,17,26,35),(9,18,27,36),(37,46,55,64),(38,47,56,65),(39,48,57,66),(40,49,58,67),(41,50,59,68),(42,51,60,69),(43,52,61,70),(44,53,62,71),(45,54,63,72),(73,100,91,82),(74,101,92,83),(75,102,93,84),(76,103,94,85),(77,104,95,86),(78,105,96,87),(79,106,97,88),(80,107,98,89),(81,108,99,90),(109,136,127,118),(110,137,128,119),(111,138,129,120),(112,139,130,121),(113,140,131,122),(114,141,132,123),(115,142,133,124),(116,143,134,125),(117,144,135,126)], [(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)], [(1,129,10,138,19,111,28,120),(2,110,11,119,20,128,29,137),(3,127,12,136,21,109,30,118),(4,144,13,117,22,126,31,135),(5,125,14,134,23,143,32,116),(6,142,15,115,24,124,33,133),(7,123,16,132,25,141,34,114),(8,140,17,113,26,122,35,131),(9,121,18,130,27,139,36,112),(37,85,46,94,55,103,64,76),(38,102,47,75,56,84,65,93),(39,83,48,92,57,101,66,74),(40,100,49,73,58,82,67,91),(41,81,50,90,59,99,68,108),(42,98,51,107,60,80,69,89),(43,79,52,88,61,97,70,106),(44,96,53,105,62,78,71,87),(45,77,54,86,63,95,72,104)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6G | 8A | ··· | 8H | 9A | 9B | 9C | 12A | ··· | 12H | 18A | ··· | 18U | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 18 | ··· | 18 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | Dic3 | M4(2) | D9 | Dic9 | D18 | Dic9 | C4.Dic3 | C4.Dic9 |
| kernel | C2×C4.Dic9 | C2×C9⋊C8 | C4.Dic9 | C22×C36 | C2×C36 | C22×C18 | C22×C12 | C2×C12 | C2×C12 | C22×C6 | C18 | C22×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 1 | 3 | 3 | 1 | 4 | 3 | 9 | 9 | 3 | 8 | 24 |
Matrix representation of C2×C4.Dic9 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 72 | 0 | 0 |
| 0 | 27 | 9 |
| 0 | 0 | 46 |
| 1 | 0 | 0 |
| 0 | 61 | 72 |
| 0 | 0 | 67 |
| 72 | 0 | 0 |
| 0 | 48 | 33 |
| 0 | 4 | 25 |
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[72,0,0,0,27,0,0,9,46],[1,0,0,0,61,0,0,72,67],[72,0,0,0,48,4,0,33,25] >;
C2×C4.Dic9 in GAP, Magma, Sage, TeX
C_2\times C_4.{\rm Dic}_9 % in TeX
G:=Group("C2xC4.Dic9"); // GroupNames label
G:=SmallGroup(288,131);
// by ID
G=gap.SmallGroup(288,131);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,422,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=1,c^18=b^2,d^2=c^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^17>;
// generators/relations