metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36.8D4, D4⋊1Dic9, C18.12D8, C18.6SD16, (D4×C9)⋊1C4, C36.7(C2×C4), (C6×D4).2S3, (C2×D4).2D9, C9⋊3(D4⋊C4), C4⋊Dic9⋊10C2, C2.3(D4⋊D9), (D4×C18).2C2, (C2×C18).34D4, (C2×C4).41D18, (C2×C12).46D6, C4.1(C2×Dic9), C6.19(D4⋊S3), C4.13(C9⋊D4), C12.6(C3⋊D4), C2.3(D4.D9), C6.9(D4.S3), C3.(D4⋊Dic3), (C3×D4).1Dic3, C12.1(C2×Dic3), (C2×C36).24C22, C18.14(C22⋊C4), C22.17(C9⋊D4), C2.4(C18.D4), C6.15(C6.D4), (C2×C9⋊C8)⋊2C2, (C2×C6).72(C3⋊D4), SmallGroup(288,40)
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=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Subgroups: 248 in 75 conjugacy classes, 36 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, C9, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×D4, C18, C18, C3⋊C8, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, D4⋊C4, Dic9, C36, C2×C18, C2×C18, C2×C3⋊C8, C4⋊Dic3, C6×D4, C9⋊C8, C2×Dic9, C2×C36, D4×C9, D4×C9, C22×C18, D4⋊Dic3, C2×C9⋊C8, C4⋊Dic9, D4×C18, D4⋊Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D8, SD16, D9, C2×Dic3, C3⋊D4, D4⋊C4, Dic9, D18, D4⋊S3, D4.S3, C6.D4, C2×Dic9, C9⋊D4, D4⋊Dic3, D4.D9, D4⋊D9, C18.D4, D4⋊Dic9
►On 144 points
Generators in S144
(1 130 43 20)(2 131 44 21)(3 132 45 22)(4 133 46 23)(5 134 47 24)(6 135 48 25)(7 136 49 26)(8 137 50 27)(9 138 51 28)(10 139 52 29)(11 140 53 30)(12 141 54 31)(13 142 37 32)(14 143 38 33)(15 144 39 34)(16 127 40 35)(17 128 41 36)(18 129 42 19)(55 85 97 120)(56 86 98 121)(57 87 99 122)(58 88 100 123)(59 89 101 124)(60 90 102 125)(61 73 103 126)(62 74 104 109)(63 75 105 110)(64 76 106 111)(65 77 107 112)(66 78 108 113)(67 79 91 114)(68 80 92 115)(69 81 93 116)(70 82 94 117)(71 83 95 118)(72 84 96 119)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 127)(50 128)(51 129)(52 130)(53 131)(54 132)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(73 117)(74 118)(75 119)(76 120)(77 121)(78 122)(79 123)(80 124)(81 125)(82 126)(83 109)(84 110)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)(99 108)
(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 88 10 79)(2 87 11 78)(3 86 12 77)(4 85 13 76)(5 84 14 75)(6 83 15 74)(7 82 16 73)(8 81 17 90)(9 80 18 89)(19 101 28 92)(20 100 29 91)(21 99 30 108)(22 98 31 107)(23 97 32 106)(24 96 33 105)(25 95 34 104)(26 94 35 103)(27 93 36 102)(37 111 46 120)(38 110 47 119)(39 109 48 118)(40 126 49 117)(41 125 50 116)(42 124 51 115)(43 123 52 114)(44 122 53 113)(45 121 54 112)(55 142 64 133)(56 141 65 132)(57 140 66 131)(58 139 67 130)(59 138 68 129)(60 137 69 128)(61 136 70 127)(62 135 71 144)(63 134 72 143)
G:=sub<Sym(144)| (1,130,43,20)(2,131,44,21)(3,132,45,22)(4,133,46,23)(5,134,47,24)(6,135,48,25)(7,136,49,26)(8,137,50,27)(9,138,51,28)(10,139,52,29)(11,140,53,30)(12,141,54,31)(13,142,37,32)(14,143,38,33)(15,144,39,34)(16,127,40,35)(17,128,41,36)(18,129,42,19)(55,85,97,120)(56,86,98,121)(57,87,99,122)(58,88,100,123)(59,89,101,124)(60,90,102,125)(61,73,103,126)(62,74,104,109)(63,75,105,110)(64,76,106,111)(65,77,107,112)(66,78,108,113)(67,79,91,114)(68,80,92,115)(69,81,93,116)(70,82,94,117)(71,83,95,118)(72,84,96,119), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,127)(50,128)(51,129)(52,130)(53,131)(54,132)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(73,117)(74,118)(75,119)(76,120)(77,121)(78,122)(79,123)(80,124)(81,125)(82,126)(83,109)(84,110)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107)(99,108), (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), (1,88,10,79)(2,87,11,78)(3,86,12,77)(4,85,13,76)(5,84,14,75)(6,83,15,74)(7,82,16,73)(8,81,17,90)(9,80,18,89)(19,101,28,92)(20,100,29,91)(21,99,30,108)(22,98,31,107)(23,97,32,106)(24,96,33,105)(25,95,34,104)(26,94,35,103)(27,93,36,102)(37,111,46,120)(38,110,47,119)(39,109,48,118)(40,126,49,117)(41,125,50,116)(42,124,51,115)(43,123,52,114)(44,122,53,113)(45,121,54,112)(55,142,64,133)(56,141,65,132)(57,140,66,131)(58,139,67,130)(59,138,68,129)(60,137,69,128)(61,136,70,127)(62,135,71,144)(63,134,72,143)>;
G:=Group( (1,130,43,20)(2,131,44,21)(3,132,45,22)(4,133,46,23)(5,134,47,24)(6,135,48,25)(7,136,49,26)(8,137,50,27)(9,138,51,28)(10,139,52,29)(11,140,53,30)(12,141,54,31)(13,142,37,32)(14,143,38,33)(15,144,39,34)(16,127,40,35)(17,128,41,36)(18,129,42,19)(55,85,97,120)(56,86,98,121)(57,87,99,122)(58,88,100,123)(59,89,101,124)(60,90,102,125)(61,73,103,126)(62,74,104,109)(63,75,105,110)(64,76,106,111)(65,77,107,112)(66,78,108,113)(67,79,91,114)(68,80,92,115)(69,81,93,116)(70,82,94,117)(71,83,95,118)(72,84,96,119), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,127)(50,128)(51,129)(52,130)(53,131)(54,132)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(73,117)(74,118)(75,119)(76,120)(77,121)(78,122)(79,123)(80,124)(81,125)(82,126)(83,109)(84,110)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107)(99,108), (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), (1,88,10,79)(2,87,11,78)(3,86,12,77)(4,85,13,76)(5,84,14,75)(6,83,15,74)(7,82,16,73)(8,81,17,90)(9,80,18,89)(19,101,28,92)(20,100,29,91)(21,99,30,108)(22,98,31,107)(23,97,32,106)(24,96,33,105)(25,95,34,104)(26,94,35,103)(27,93,36,102)(37,111,46,120)(38,110,47,119)(39,109,48,118)(40,126,49,117)(41,125,50,116)(42,124,51,115)(43,123,52,114)(44,122,53,113)(45,121,54,112)(55,142,64,133)(56,141,65,132)(57,140,66,131)(58,139,67,130)(59,138,68,129)(60,137,69,128)(61,136,70,127)(62,135,71,144)(63,134,72,143) );
G=PermutationGroup([[(1,130,43,20),(2,131,44,21),(3,132,45,22),(4,133,46,23),(5,134,47,24),(6,135,48,25),(7,136,49,26),(8,137,50,27),(9,138,51,28),(10,139,52,29),(11,140,53,30),(12,141,54,31),(13,142,37,32),(14,143,38,33),(15,144,39,34),(16,127,40,35),(17,128,41,36),(18,129,42,19),(55,85,97,120),(56,86,98,121),(57,87,99,122),(58,88,100,123),(59,89,101,124),(60,90,102,125),(61,73,103,126),(62,74,104,109),(63,75,105,110),(64,76,106,111),(65,77,107,112),(66,78,108,113),(67,79,91,114),(68,80,92,115),(69,81,93,116),(70,82,94,117),(71,83,95,118),(72,84,96,119)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(37,133),(38,134),(39,135),(40,136),(41,137),(42,138),(43,139),(44,140),(45,141),(46,142),(47,143),(48,144),(49,127),(50,128),(51,129),(52,130),(53,131),(54,132),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72),(73,117),(74,118),(75,119),(76,120),(77,121),(78,122),(79,123),(80,124),(81,125),(82,126),(83,109),(84,110),(85,111),(86,112),(87,113),(88,114),(89,115),(90,116),(91,100),(92,101),(93,102),(94,103),(95,104),(96,105),(97,106),(98,107),(99,108)], [(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)], [(1,88,10,79),(2,87,11,78),(3,86,12,77),(4,85,13,76),(5,84,14,75),(6,83,15,74),(7,82,16,73),(8,81,17,90),(9,80,18,89),(19,101,28,92),(20,100,29,91),(21,99,30,108),(22,98,31,107),(23,97,32,106),(24,96,33,105),(25,95,34,104),(26,94,35,103),(27,93,36,102),(37,111,46,120),(38,110,47,119),(39,109,48,118),(40,126,49,117),(41,125,50,116),(42,124,51,115),(43,123,52,114),(44,122,53,113),(45,121,54,112),(55,142,64,133),(56,141,65,132),(57,140,66,131),(58,139,67,130),(59,138,68,129),(60,137,69,128),(61,136,70,127),(62,135,71,144),(63,134,72,143)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 18A | ··· | 18I | 18J | ··· | 18U | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | Dic3 | D8 | SD16 | D9 | C3⋊D4 | C3⋊D4 | D18 | Dic9 | C9⋊D4 | C9⋊D4 | D4⋊S3 | D4.S3 | D4.D9 | D4⋊D9 |
| kernel | D4⋊Dic9 | C2×C9⋊C8 | C4⋊Dic9 | D4×C18 | D4×C9 | C6×D4 | C36 | C2×C18 | C2×C12 | C3×D4 | C18 | C18 | C2×D4 | C12 | C2×C6 | C2×C4 | D4 | C4 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 1 | 1 | 3 | 3 |
Matrix representation of D4⋊Dic9 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 29 |
| 0 | 0 | 10 | 72 |
| 1 | 0 | 0 | 0 |
| 16 | 72 | 0 | 0 |
| 0 | 0 | 1 | 29 |
| 0 | 0 | 0 | 72 |
| 36 | 0 | 0 | 0 |
| 12 | 71 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 18 | 0 | 0 |
| 20 | 71 | 0 | 0 |
| 0 | 0 | 0 | 26 |
| 0 | 0 | 59 | 0 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,10,0,0,29,72],[1,16,0,0,0,72,0,0,0,0,1,0,0,0,29,72],[36,12,0,0,0,71,0,0,0,0,1,0,0,0,0,1],[2,20,0,0,18,71,0,0,0,0,0,59,0,0,26,0] >;
D4⋊Dic9 in GAP, Magma, Sage, TeX
D_4\rtimes {\rm Dic}_9 % in TeX
G:=Group("D4:Dic9"); // GroupNames label
G:=SmallGroup(288,40);
// by ID
G=gap.SmallGroup(288,40);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,675,346,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^18=1,d^2=c^9,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations