direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×Dic9, C23.22D18, C9⋊5(C4×D4), C36⋊3(C2×C4), (D4×C9)⋊3C4, C2.5(D4×D9), C3.(D4×Dic3), (C6×D4).6S3, (C2×D4).7D9, C4⋊1(C2×Dic9), C6.99(S3×D4), C4⋊Dic9⋊13C2, (C4×Dic9)⋊4C2, (D4×C18).4C2, (C2×C12).58D6, C18.37(C2×D4), (C2×C4).50D18, (C3×D4).3Dic3, C12.4(C2×Dic3), C22⋊2(C2×Dic9), (C22×C6).47D6, C18.28(C4○D4), C2.5(D4⋊2D9), C18.D4⋊7C2, C18.25(C22×C4), (C2×C18).49C23, (C2×C36).36C22, (C22×Dic9)⋊4C2, C6.85(D4⋊2S3), C2.6(C22×Dic9), C6.26(C22×Dic3), C22.25(C22×D9), (C22×C18).17C22, (C2×Dic9).49C22, (C2×C18)⋊3(C2×C4), (C2×C6).2(C2×Dic3), (C2×C6).206(C22×S3), SmallGroup(288,144)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×Dic9
G = < a,b,c,d | a4=b2=c18=1, d2=c9, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 456 in 141 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C9, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C18, C18, C2×Dic3, C2×C12, C3×D4, C22×C6, C4×D4, Dic9, Dic9, C36, C2×C18, C2×C18, C2×C18, C4×Dic3, C4⋊Dic3, C6.D4, C22×Dic3, C6×D4, C2×Dic9, C2×Dic9, C2×Dic9, C2×C36, D4×C9, C22×C18, D4×Dic3, C4×Dic9, C4⋊Dic9, C18.D4, C22×Dic9, D4×C18, D4×Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22×C4, C2×D4, C4○D4, D9, C2×Dic3, C22×S3, C4×D4, Dic9, D18, S3×D4, D4⋊2S3, C22×Dic3, C2×Dic9, C22×D9, D4×Dic3, D4×D9, D4⋊2D9, C22×Dic9, D4×Dic9
►On 144 points
Generators in S144
(1 38 141 123)(2 39 142 124)(3 40 143 125)(4 41 144 126)(5 42 127 109)(6 43 128 110)(7 44 129 111)(8 45 130 112)(9 46 131 113)(10 47 132 114)(11 48 133 115)(12 49 134 116)(13 50 135 117)(14 51 136 118)(15 52 137 119)(16 53 138 120)(17 54 139 121)(18 37 140 122)(19 57 105 84)(20 58 106 85)(21 59 107 86)(22 60 108 87)(23 61 91 88)(24 62 92 89)(25 63 93 90)(26 64 94 73)(27 65 95 74)(28 66 96 75)(29 67 97 76)(30 68 98 77)(31 69 99 78)(32 70 100 79)(33 71 101 80)(34 72 102 81)(35 55 103 82)(36 56 104 83)
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 109)(15 110)(16 111)(17 112)(18 113)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 72)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 131)(38 132)(39 133)(40 134)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 141)(48 142)(49 143)(50 144)(51 127)(52 128)(53 129)(54 130)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)(85 97)(86 98)(87 99)(88 100)(89 101)(90 102)
(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 72 10 63)(2 71 11 62)(3 70 12 61)(4 69 13 60)(5 68 14 59)(6 67 15 58)(7 66 16 57)(8 65 17 56)(9 64 18 55)(19 111 28 120)(20 110 29 119)(21 109 30 118)(22 126 31 117)(23 125 32 116)(24 124 33 115)(25 123 34 114)(26 122 35 113)(27 121 36 112)(37 103 46 94)(38 102 47 93)(39 101 48 92)(40 100 49 91)(41 99 50 108)(42 98 51 107)(43 97 52 106)(44 96 53 105)(45 95 54 104)(73 140 82 131)(74 139 83 130)(75 138 84 129)(76 137 85 128)(77 136 86 127)(78 135 87 144)(79 134 88 143)(80 133 89 142)(81 132 90 141)
G:=sub<Sym(144)| (1,38,141,123)(2,39,142,124)(3,40,143,125)(4,41,144,126)(5,42,127,109)(6,43,128,110)(7,44,129,111)(8,45,130,112)(9,46,131,113)(10,47,132,114)(11,48,133,115)(12,49,134,116)(13,50,135,117)(14,51,136,118)(15,52,137,119)(16,53,138,120)(17,54,139,121)(18,37,140,122)(19,57,105,84)(20,58,106,85)(21,59,107,86)(22,60,108,87)(23,61,91,88)(24,62,92,89)(25,63,93,90)(26,64,94,73)(27,65,95,74)(28,66,96,75)(29,67,97,76)(30,68,98,77)(31,69,99,78)(32,70,100,79)(33,71,101,80)(34,72,102,81)(35,55,103,82)(36,56,104,83), (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,109)(15,110)(16,111)(17,112)(18,113)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,131)(38,132)(39,133)(40,134)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,143)(50,144)(51,127)(52,128)(53,129)(54,130)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102), (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), (1,72,10,63)(2,71,11,62)(3,70,12,61)(4,69,13,60)(5,68,14,59)(6,67,15,58)(7,66,16,57)(8,65,17,56)(9,64,18,55)(19,111,28,120)(20,110,29,119)(21,109,30,118)(22,126,31,117)(23,125,32,116)(24,124,33,115)(25,123,34,114)(26,122,35,113)(27,121,36,112)(37,103,46,94)(38,102,47,93)(39,101,48,92)(40,100,49,91)(41,99,50,108)(42,98,51,107)(43,97,52,106)(44,96,53,105)(45,95,54,104)(73,140,82,131)(74,139,83,130)(75,138,84,129)(76,137,85,128)(77,136,86,127)(78,135,87,144)(79,134,88,143)(80,133,89,142)(81,132,90,141)>;
G:=Group( (1,38,141,123)(2,39,142,124)(3,40,143,125)(4,41,144,126)(5,42,127,109)(6,43,128,110)(7,44,129,111)(8,45,130,112)(9,46,131,113)(10,47,132,114)(11,48,133,115)(12,49,134,116)(13,50,135,117)(14,51,136,118)(15,52,137,119)(16,53,138,120)(17,54,139,121)(18,37,140,122)(19,57,105,84)(20,58,106,85)(21,59,107,86)(22,60,108,87)(23,61,91,88)(24,62,92,89)(25,63,93,90)(26,64,94,73)(27,65,95,74)(28,66,96,75)(29,67,97,76)(30,68,98,77)(31,69,99,78)(32,70,100,79)(33,71,101,80)(34,72,102,81)(35,55,103,82)(36,56,104,83), (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,109)(15,110)(16,111)(17,112)(18,113)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,131)(38,132)(39,133)(40,134)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,143)(50,144)(51,127)(52,128)(53,129)(54,130)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102), (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), (1,72,10,63)(2,71,11,62)(3,70,12,61)(4,69,13,60)(5,68,14,59)(6,67,15,58)(7,66,16,57)(8,65,17,56)(9,64,18,55)(19,111,28,120)(20,110,29,119)(21,109,30,118)(22,126,31,117)(23,125,32,116)(24,124,33,115)(25,123,34,114)(26,122,35,113)(27,121,36,112)(37,103,46,94)(38,102,47,93)(39,101,48,92)(40,100,49,91)(41,99,50,108)(42,98,51,107)(43,97,52,106)(44,96,53,105)(45,95,54,104)(73,140,82,131)(74,139,83,130)(75,138,84,129)(76,137,85,128)(77,136,86,127)(78,135,87,144)(79,134,88,143)(80,133,89,142)(81,132,90,141) );
G=PermutationGroup([[(1,38,141,123),(2,39,142,124),(3,40,143,125),(4,41,144,126),(5,42,127,109),(6,43,128,110),(7,44,129,111),(8,45,130,112),(9,46,131,113),(10,47,132,114),(11,48,133,115),(12,49,134,116),(13,50,135,117),(14,51,136,118),(15,52,137,119),(16,53,138,120),(17,54,139,121),(18,37,140,122),(19,57,105,84),(20,58,106,85),(21,59,107,86),(22,60,108,87),(23,61,91,88),(24,62,92,89),(25,63,93,90),(26,64,94,73),(27,65,95,74),(28,66,96,75),(29,67,97,76),(30,68,98,77),(31,69,99,78),(32,70,100,79),(33,71,101,80),(34,72,102,81),(35,55,103,82),(36,56,104,83)], [(1,114),(2,115),(3,116),(4,117),(5,118),(6,119),(7,120),(8,121),(9,122),(10,123),(11,124),(12,125),(13,126),(14,109),(15,110),(16,111),(17,112),(18,113),(19,66),(20,67),(21,68),(22,69),(23,70),(24,71),(25,72),(26,55),(27,56),(28,57),(29,58),(30,59),(31,60),(32,61),(33,62),(34,63),(35,64),(36,65),(37,131),(38,132),(39,133),(40,134),(41,135),(42,136),(43,137),(44,138),(45,139),(46,140),(47,141),(48,142),(49,143),(50,144),(51,127),(52,128),(53,129),(54,130),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96),(85,97),(86,98),(87,99),(88,100),(89,101),(90,102)], [(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)], [(1,72,10,63),(2,71,11,62),(3,70,12,61),(4,69,13,60),(5,68,14,59),(6,67,15,58),(7,66,16,57),(8,65,17,56),(9,64,18,55),(19,111,28,120),(20,110,29,119),(21,109,30,118),(22,126,31,117),(23,125,32,116),(24,124,33,115),(25,123,34,114),(26,122,35,113),(27,121,36,112),(37,103,46,94),(38,102,47,93),(39,101,48,92),(40,100,49,91),(41,99,50,108),(42,98,51,107),(43,97,52,106),(44,96,53,105),(45,95,54,104),(73,140,82,131),(74,139,83,130),(75,138,84,129),(76,137,85,128),(77,136,86,127),(78,135,87,144),(79,134,88,143),(80,133,89,142),(81,132,90,141)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 12A | 12B | 18A | ··· | 18I | 18J | ··· | 18U | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | Dic3 | D6 | C4○D4 | D9 | D18 | Dic9 | D18 | S3×D4 | D4⋊2S3 | D4×D9 | D4⋊2D9 |
| kernel | D4×Dic9 | C4×Dic9 | C4⋊Dic9 | C18.D4 | C22×Dic9 | D4×C18 | D4×C9 | C6×D4 | Dic9 | C2×C12 | C3×D4 | C22×C6 | C18 | C2×D4 | C2×C4 | D4 | C23 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 8 | 1 | 2 | 1 | 4 | 2 | 2 | 3 | 3 | 12 | 6 | 1 | 1 | 3 | 3 |
Matrix representation of D4×Dic9 ►in GL5(𝔽37)
| 36 | 0 | 0 | 0 | 0 |
| 0 | 1 | 35 | 0 | 0 |
| 0 | 1 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 | 0 |
| 0 | 1 | 35 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 | 20 |
| 0 | 0 | 0 | 17 | 31 |
| 6 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 | 1 |
G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,1,1,0,0,0,35,36,0,0,0,0,0,36,0,0,0,0,0,36],[36,0,0,0,0,0,1,0,0,0,0,35,36,0,0,0,0,0,36,0,0,0,0,0,36],[36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,17,0,0,0,20,31],[6,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36,36,0,0,0,0,1] >;
D4×Dic9 in GAP, Magma, Sage, TeX
D_4\times {\rm Dic}_9 % in TeX
G:=Group("D4xDic9"); // GroupNames label
G:=SmallGroup(288,144);
// by ID
G=gap.SmallGroup(288,144);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,219,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^18=1,d^2=c^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations