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 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×4], C22 [×4], C6 [×3], C6 [×4], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C9, Dic3 [×5], C12 [×2], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C18 [×3], C18 [×4], C2×Dic3 [×8], C2×C12, C3×D4 [×4], C22×C6 [×2], C4×D4, Dic9 [×2], Dic9 [×3], C36 [×2], C2×C18, C2×C18 [×4], C2×C18 [×4], C4×Dic3, C4⋊Dic3, C6.D4 [×2], C22×Dic3 [×2], C6×D4, C2×Dic9 [×2], C2×Dic9 [×2], C2×Dic9 [×4], C2×C36, D4×C9 [×4], C22×C18 [×2], D4×Dic3, C4×Dic9, C4⋊Dic9, C18.D4 [×2], C22×Dic9 [×2], D4×C18, D4×Dic9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, Dic3 [×4], D6 [×3], C22×C4, C2×D4, C4○D4, D9, C2×Dic3 [×6], C22×S3, C4×D4, Dic9 [×4], D18 [×3], S3×D4, D4⋊2S3, C22×Dic3, C2×Dic9 [×6], C22×D9, D4×Dic3, D4×D9, D4⋊2D9, C22×Dic9, D4×Dic9
►On 144 points
Generators in S144
(1 131 105 30)(2 132 106 31)(3 133 107 32)(4 134 108 33)(5 135 91 34)(6 136 92 35)(7 137 93 36)(8 138 94 19)(9 139 95 20)(10 140 96 21)(11 141 97 22)(12 142 98 23)(13 143 99 24)(14 144 100 25)(15 127 101 26)(16 128 102 27)(17 129 103 28)(18 130 104 29)(37 84 117 58)(38 85 118 59)(39 86 119 60)(40 87 120 61)(41 88 121 62)(42 89 122 63)(43 90 123 64)(44 73 124 65)(45 74 125 66)(46 75 126 67)(47 76 109 68)(48 77 110 69)(49 78 111 70)(50 79 112 71)(51 80 113 72)(52 81 114 55)(53 82 115 56)(54 83 116 57)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 19)(18 20)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 73)(54 74)(55 123)(56 124)(57 125)(58 126)(59 109)(60 110)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 121)(72 122)(91 144)(92 127)(93 128)(94 129)(95 130)(96 131)(97 132)(98 133)(99 134)(100 135)(101 136)(102 137)(103 138)(104 139)(105 140)(106 141)(107 142)(108 143)
(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 56 10 65)(2 55 11 64)(3 72 12 63)(4 71 13 62)(5 70 14 61)(6 69 15 60)(7 68 16 59)(8 67 17 58)(9 66 18 57)(19 126 28 117)(20 125 29 116)(21 124 30 115)(22 123 31 114)(23 122 32 113)(24 121 33 112)(25 120 34 111)(26 119 35 110)(27 118 36 109)(37 138 46 129)(38 137 47 128)(39 136 48 127)(40 135 49 144)(41 134 50 143)(42 133 51 142)(43 132 52 141)(44 131 53 140)(45 130 54 139)(73 105 82 96)(74 104 83 95)(75 103 84 94)(76 102 85 93)(77 101 86 92)(78 100 87 91)(79 99 88 108)(80 98 89 107)(81 97 90 106)
G:=sub<Sym(144)| (1,131,105,30)(2,132,106,31)(3,133,107,32)(4,134,108,33)(5,135,91,34)(6,136,92,35)(7,137,93,36)(8,138,94,19)(9,139,95,20)(10,140,96,21)(11,141,97,22)(12,142,98,23)(13,143,99,24)(14,144,100,25)(15,127,101,26)(16,128,102,27)(17,129,103,28)(18,130,104,29)(37,84,117,58)(38,85,118,59)(39,86,119,60)(40,87,120,61)(41,88,121,62)(42,89,122,63)(43,90,123,64)(44,73,124,65)(45,74,125,66)(46,75,126,67)(47,76,109,68)(48,77,110,69)(49,78,111,70)(50,79,112,71)(51,80,113,72)(52,81,114,55)(53,82,115,56)(54,83,116,57), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,19)(18,20)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,73)(54,74)(55,123)(56,124)(57,125)(58,126)(59,109)(60,110)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,121)(72,122)(91,144)(92,127)(93,128)(94,129)(95,130)(96,131)(97,132)(98,133)(99,134)(100,135)(101,136)(102,137)(103,138)(104,139)(105,140)(106,141)(107,142)(108,143), (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), (1,56,10,65)(2,55,11,64)(3,72,12,63)(4,71,13,62)(5,70,14,61)(6,69,15,60)(7,68,16,59)(8,67,17,58)(9,66,18,57)(19,126,28,117)(20,125,29,116)(21,124,30,115)(22,123,31,114)(23,122,32,113)(24,121,33,112)(25,120,34,111)(26,119,35,110)(27,118,36,109)(37,138,46,129)(38,137,47,128)(39,136,48,127)(40,135,49,144)(41,134,50,143)(42,133,51,142)(43,132,52,141)(44,131,53,140)(45,130,54,139)(73,105,82,96)(74,104,83,95)(75,103,84,94)(76,102,85,93)(77,101,86,92)(78,100,87,91)(79,99,88,108)(80,98,89,107)(81,97,90,106)>;
G:=Group( (1,131,105,30)(2,132,106,31)(3,133,107,32)(4,134,108,33)(5,135,91,34)(6,136,92,35)(7,137,93,36)(8,138,94,19)(9,139,95,20)(10,140,96,21)(11,141,97,22)(12,142,98,23)(13,143,99,24)(14,144,100,25)(15,127,101,26)(16,128,102,27)(17,129,103,28)(18,130,104,29)(37,84,117,58)(38,85,118,59)(39,86,119,60)(40,87,120,61)(41,88,121,62)(42,89,122,63)(43,90,123,64)(44,73,124,65)(45,74,125,66)(46,75,126,67)(47,76,109,68)(48,77,110,69)(49,78,111,70)(50,79,112,71)(51,80,113,72)(52,81,114,55)(53,82,115,56)(54,83,116,57), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,19)(18,20)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,73)(54,74)(55,123)(56,124)(57,125)(58,126)(59,109)(60,110)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,121)(72,122)(91,144)(92,127)(93,128)(94,129)(95,130)(96,131)(97,132)(98,133)(99,134)(100,135)(101,136)(102,137)(103,138)(104,139)(105,140)(106,141)(107,142)(108,143), (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), (1,56,10,65)(2,55,11,64)(3,72,12,63)(4,71,13,62)(5,70,14,61)(6,69,15,60)(7,68,16,59)(8,67,17,58)(9,66,18,57)(19,126,28,117)(20,125,29,116)(21,124,30,115)(22,123,31,114)(23,122,32,113)(24,121,33,112)(25,120,34,111)(26,119,35,110)(27,118,36,109)(37,138,46,129)(38,137,47,128)(39,136,48,127)(40,135,49,144)(41,134,50,143)(42,133,51,142)(43,132,52,141)(44,131,53,140)(45,130,54,139)(73,105,82,96)(74,104,83,95)(75,103,84,94)(76,102,85,93)(77,101,86,92)(78,100,87,91)(79,99,88,108)(80,98,89,107)(81,97,90,106) );
G=PermutationGroup([(1,131,105,30),(2,132,106,31),(3,133,107,32),(4,134,108,33),(5,135,91,34),(6,136,92,35),(7,137,93,36),(8,138,94,19),(9,139,95,20),(10,140,96,21),(11,141,97,22),(12,142,98,23),(13,143,99,24),(14,144,100,25),(15,127,101,26),(16,128,102,27),(17,129,103,28),(18,130,104,29),(37,84,117,58),(38,85,118,59),(39,86,119,60),(40,87,120,61),(41,88,121,62),(42,89,122,63),(43,90,123,64),(44,73,124,65),(45,74,125,66),(46,75,126,67),(47,76,109,68),(48,77,110,69),(49,78,111,70),(50,79,112,71),(51,80,113,72),(52,81,114,55),(53,82,115,56),(54,83,116,57)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,19),(18,20),(37,75),(38,76),(39,77),(40,78),(41,79),(42,80),(43,81),(44,82),(45,83),(46,84),(47,85),(48,86),(49,87),(50,88),(51,89),(52,90),(53,73),(54,74),(55,123),(56,124),(57,125),(58,126),(59,109),(60,110),(61,111),(62,112),(63,113),(64,114),(65,115),(66,116),(67,117),(68,118),(69,119),(70,120),(71,121),(72,122),(91,144),(92,127),(93,128),(94,129),(95,130),(96,131),(97,132),(98,133),(99,134),(100,135),(101,136),(102,137),(103,138),(104,139),(105,140),(106,141),(107,142),(108,143)], [(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)], [(1,56,10,65),(2,55,11,64),(3,72,12,63),(4,71,13,62),(5,70,14,61),(6,69,15,60),(7,68,16,59),(8,67,17,58),(9,66,18,57),(19,126,28,117),(20,125,29,116),(21,124,30,115),(22,123,31,114),(23,122,32,113),(24,121,33,112),(25,120,34,111),(26,119,35,110),(27,118,36,109),(37,138,46,129),(38,137,47,128),(39,136,48,127),(40,135,49,144),(41,134,50,143),(42,133,51,142),(43,132,52,141),(44,131,53,140),(45,130,54,139),(73,105,82,96),(74,104,83,95),(75,103,84,94),(76,102,85,93),(77,101,86,92),(78,100,87,91),(79,99,88,108),(80,98,89,107),(81,97,90,106)])
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