direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8⋊3D9, Q8⋊6D18, D36⋊9C22, C18.9C24, C36.23C23, D18.4C23, Dic9.9C23, (C2×Q8)⋊8D9, (Q8×C18)⋊6C2, (C2×D36)⋊12C2, C18⋊3(C4○D4), (C2×C4).62D18, (C4×D9)⋊5C22, (C3×Q8).60D6, (C6×Q8).21S3, (Q8×C9)⋊6C22, (C2×C12).102D6, C6.46(S3×C23), C2.10(C23×D9), C4.23(C22×D9), (C2×C18).67C23, C12.63(C22×S3), (C2×C36).50C22, C6.46(Q8⋊3S3), C22.32(C22×D9), (C2×Dic9).51C22, (C22×D9).30C22, (C2×C4×D9)⋊5C2, C9⋊3(C2×C4○D4), C3.(C2×Q8⋊3S3), (C2×C6).225(C22×S3), SmallGroup(288,360)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8⋊3D9
G = < a,b,c,d,e | a2=b4=d9=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1032 in 246 conjugacy classes, 108 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C9, Dic3, C12, D6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, D9, C18, C18, C4×S3, D12, C2×Dic3, C2×C12, C3×Q8, C22×S3, C2×C4○D4, Dic9, C36, D18, D18, C2×C18, S3×C2×C4, C2×D12, Q8⋊3S3, C6×Q8, C4×D9, D36, C2×Dic9, C2×C36, Q8×C9, C22×D9, C2×Q8⋊3S3, C2×C4×D9, C2×D36, Q8⋊3D9, Q8×C18, C2×Q8⋊3D9
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, D9, C22×S3, C2×C4○D4, D18, Q8⋊3S3, S3×C23, C22×D9, C2×Q8⋊3S3, Q8⋊3D9, C23×D9, C2×Q8⋊3D9
►On 144 points
Generators in S144
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(1 64 10 55)(2 65 11 56)(3 66 12 57)(4 67 13 58)(5 68 14 59)(6 69 15 60)(7 70 16 61)(8 71 17 62)(9 72 18 63)(19 37 28 46)(20 38 29 47)(21 39 30 48)(22 40 31 49)(23 41 32 50)(24 42 33 51)(25 43 34 52)(26 44 35 53)(27 45 36 54)(73 136 82 127)(74 137 83 128)(75 138 84 129)(76 139 85 130)(77 140 86 131)(78 141 87 132)(79 142 88 133)(80 143 89 134)(81 144 90 135)(91 109 100 118)(92 110 101 119)(93 111 102 120)(94 112 103 121)(95 113 104 122)(96 114 105 123)(97 115 106 124)(98 116 107 125)(99 117 108 126)
(1 109 10 118)(2 110 11 119)(3 111 12 120)(4 112 13 121)(5 113 14 122)(6 114 15 123)(7 115 16 124)(8 116 17 125)(9 117 18 126)(19 127 28 136)(20 128 29 137)(21 129 30 138)(22 130 31 139)(23 131 32 140)(24 132 33 141)(25 133 34 142)(26 134 35 143)(27 135 36 144)(37 82 46 73)(38 83 47 74)(39 84 48 75)(40 85 49 76)(41 86 50 77)(42 87 51 78)(43 88 52 79)(44 89 53 80)(45 90 54 81)(55 100 64 91)(56 101 65 92)(57 102 66 93)(58 103 67 94)(59 104 68 95)(60 105 69 96)(61 106 70 97)(62 107 71 98)(63 108 72 99)
(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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 72)(11 71)(12 70)(13 69)(14 68)(15 67)(16 66)(17 65)(18 64)(19 54)(20 53)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(73 135)(74 134)(75 133)(76 132)(77 131)(78 130)(79 129)(80 128)(81 127)(82 144)(83 143)(84 142)(85 141)(86 140)(87 139)(88 138)(89 137)(90 136)(91 126)(92 125)(93 124)(94 123)(95 122)(96 121)(97 120)(98 119)(99 118)(100 117)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)
G:=sub<Sym(144)| (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (1,64,10,55)(2,65,11,56)(3,66,12,57)(4,67,13,58)(5,68,14,59)(6,69,15,60)(7,70,16,61)(8,71,17,62)(9,72,18,63)(19,37,28,46)(20,38,29,47)(21,39,30,48)(22,40,31,49)(23,41,32,50)(24,42,33,51)(25,43,34,52)(26,44,35,53)(27,45,36,54)(73,136,82,127)(74,137,83,128)(75,138,84,129)(76,139,85,130)(77,140,86,131)(78,141,87,132)(79,142,88,133)(80,143,89,134)(81,144,90,135)(91,109,100,118)(92,110,101,119)(93,111,102,120)(94,112,103,121)(95,113,104,122)(96,114,105,123)(97,115,106,124)(98,116,107,125)(99,117,108,126), (1,109,10,118)(2,110,11,119)(3,111,12,120)(4,112,13,121)(5,113,14,122)(6,114,15,123)(7,115,16,124)(8,116,17,125)(9,117,18,126)(19,127,28,136)(20,128,29,137)(21,129,30,138)(22,130,31,139)(23,131,32,140)(24,132,33,141)(25,133,34,142)(26,134,35,143)(27,135,36,144)(37,82,46,73)(38,83,47,74)(39,84,48,75)(40,85,49,76)(41,86,50,77)(42,87,51,78)(43,88,52,79)(44,89,53,80)(45,90,54,81)(55,100,64,91)(56,101,65,92)(57,102,66,93)(58,103,67,94)(59,104,68,95)(60,105,69,96)(61,106,70,97)(62,107,71,98)(63,108,72,99), (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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,72)(11,71)(12,70)(13,69)(14,68)(15,67)(16,66)(17,65)(18,64)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(73,135)(74,134)(75,133)(76,132)(77,131)(78,130)(79,129)(80,128)(81,127)(82,144)(83,143)(84,142)(85,141)(86,140)(87,139)(88,138)(89,137)(90,136)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)>;
G:=Group( (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (1,64,10,55)(2,65,11,56)(3,66,12,57)(4,67,13,58)(5,68,14,59)(6,69,15,60)(7,70,16,61)(8,71,17,62)(9,72,18,63)(19,37,28,46)(20,38,29,47)(21,39,30,48)(22,40,31,49)(23,41,32,50)(24,42,33,51)(25,43,34,52)(26,44,35,53)(27,45,36,54)(73,136,82,127)(74,137,83,128)(75,138,84,129)(76,139,85,130)(77,140,86,131)(78,141,87,132)(79,142,88,133)(80,143,89,134)(81,144,90,135)(91,109,100,118)(92,110,101,119)(93,111,102,120)(94,112,103,121)(95,113,104,122)(96,114,105,123)(97,115,106,124)(98,116,107,125)(99,117,108,126), (1,109,10,118)(2,110,11,119)(3,111,12,120)(4,112,13,121)(5,113,14,122)(6,114,15,123)(7,115,16,124)(8,116,17,125)(9,117,18,126)(19,127,28,136)(20,128,29,137)(21,129,30,138)(22,130,31,139)(23,131,32,140)(24,132,33,141)(25,133,34,142)(26,134,35,143)(27,135,36,144)(37,82,46,73)(38,83,47,74)(39,84,48,75)(40,85,49,76)(41,86,50,77)(42,87,51,78)(43,88,52,79)(44,89,53,80)(45,90,54,81)(55,100,64,91)(56,101,65,92)(57,102,66,93)(58,103,67,94)(59,104,68,95)(60,105,69,96)(61,106,70,97)(62,107,71,98)(63,108,72,99), (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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,72)(11,71)(12,70)(13,69)(14,68)(15,67)(16,66)(17,65)(18,64)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(73,135)(74,134)(75,133)(76,132)(77,131)(78,130)(79,129)(80,128)(81,127)(82,144)(83,143)(84,142)(85,141)(86,140)(87,139)(88,138)(89,137)(90,136)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109) );
G=PermutationGroup([[(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(49,121),(50,122),(51,123),(52,124),(53,125),(54,126),(55,127),(56,128),(57,129),(58,130),(59,131),(60,132),(61,133),(62,134),(63,135),(64,136),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144)], [(1,64,10,55),(2,65,11,56),(3,66,12,57),(4,67,13,58),(5,68,14,59),(6,69,15,60),(7,70,16,61),(8,71,17,62),(9,72,18,63),(19,37,28,46),(20,38,29,47),(21,39,30,48),(22,40,31,49),(23,41,32,50),(24,42,33,51),(25,43,34,52),(26,44,35,53),(27,45,36,54),(73,136,82,127),(74,137,83,128),(75,138,84,129),(76,139,85,130),(77,140,86,131),(78,141,87,132),(79,142,88,133),(80,143,89,134),(81,144,90,135),(91,109,100,118),(92,110,101,119),(93,111,102,120),(94,112,103,121),(95,113,104,122),(96,114,105,123),(97,115,106,124),(98,116,107,125),(99,117,108,126)], [(1,109,10,118),(2,110,11,119),(3,111,12,120),(4,112,13,121),(5,113,14,122),(6,114,15,123),(7,115,16,124),(8,116,17,125),(9,117,18,126),(19,127,28,136),(20,128,29,137),(21,129,30,138),(22,130,31,139),(23,131,32,140),(24,132,33,141),(25,133,34,142),(26,134,35,143),(27,135,36,144),(37,82,46,73),(38,83,47,74),(39,84,48,75),(40,85,49,76),(41,86,50,77),(42,87,51,78),(43,88,52,79),(44,89,53,80),(45,90,54,81),(55,100,64,91),(56,101,65,92),(57,102,66,93),(58,103,67,94),(59,104,68,95),(60,105,69,96),(61,106,70,97),(62,107,71,98),(63,108,72,99)], [(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)], [(1,63),(2,62),(3,61),(4,60),(5,59),(6,58),(7,57),(8,56),(9,55),(10,72),(11,71),(12,70),(13,69),(14,68),(15,67),(16,66),(17,65),(18,64),(19,54),(20,53),(21,52),(22,51),(23,50),(24,49),(25,48),(26,47),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37),(73,135),(74,134),(75,133),(76,132),(77,131),(78,130),(79,129),(80,128),(81,127),(82,144),(83,143),(84,142),(85,141),(86,140),(87,139),(88,138),(89,137),(90,136),(91,126),(92,125),(93,124),(94,123),(95,122),(96,121),(97,120),(98,119),(99,118),(100,117),(101,116),(102,115),(103,114),(104,113),(105,112),(106,111),(107,110),(108,109)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 18 | ··· | 18 | 2 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | D9 | D18 | D18 | Q8⋊3S3 | Q8⋊3D9 |
| kernel | C2×Q8⋊3D9 | C2×C4×D9 | C2×D36 | Q8⋊3D9 | Q8×C18 | C6×Q8 | C2×C12 | C3×Q8 | C18 | C2×Q8 | C2×C4 | Q8 | C6 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 1 | 3 | 4 | 4 | 3 | 9 | 12 | 2 | 6 |
Matrix representation of C2×Q8⋊3D9 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 29 | 12 |
| 0 | 0 | 10 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 2 |
| 0 | 0 | 13 | 26 |
| 6 | 26 | 0 | 0 |
| 11 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 31 | 11 | 0 | 0 |
| 17 | 6 | 0 | 0 |
| 0 | 0 | 29 | 12 |
| 0 | 0 | 4 | 8 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,29,10,0,0,12,8],[1,0,0,0,0,1,0,0,0,0,11,13,0,0,2,26],[6,11,0,0,26,17,0,0,0,0,1,0,0,0,0,1],[31,17,0,0,11,6,0,0,0,0,29,4,0,0,12,8] >;
C2×Q8⋊3D9 in GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes_3D_9
% in TeX
G:=Group("C2xQ8:3D9"); // GroupNames label
G:=SmallGroup(288,360);
// by ID
G=gap.SmallGroup(288,360);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,185,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^9=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations