metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic9⋊C4, C18.5D4, C18.1Q8, C6.3Dic6, C2.1Dic18, C22.4D18, C9⋊1(C4⋊C4), C6.8(C4×S3), (C2×C4).1D9, C2.4(C4×D9), C18.4(C2×C4), (C2×C36).1C2, (C2×C12).1S3, (C2×C6).20D6, C2.1(C9⋊D4), C3.(Dic3⋊C4), C6.12(C3⋊D4), (C2×C18).4C22, (C2×Dic9).1C2, SmallGroup(144,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic9⋊C4
G = < a,b,c | a18=c4=1, b2=a9, bab-1=a-1, ac=ca, cbc-1=a9b >
Smallest permutation representation of Dic9⋊C4
►Regular action on 144 points
Generators in S144
(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 74 10 83)(2 73 11 82)(3 90 12 81)(4 89 13 80)(5 88 14 79)(6 87 15 78)(7 86 16 77)(8 85 17 76)(9 84 18 75)(19 103 28 94)(20 102 29 93)(21 101 30 92)(22 100 31 91)(23 99 32 108)(24 98 33 107)(25 97 34 106)(26 96 35 105)(27 95 36 104)(37 125 46 116)(38 124 47 115)(39 123 48 114)(40 122 49 113)(41 121 50 112)(42 120 51 111)(43 119 52 110)(44 118 53 109)(45 117 54 126)(55 144 64 135)(56 143 65 134)(57 142 66 133)(58 141 67 132)(59 140 68 131)(60 139 69 130)(61 138 70 129)(62 137 71 128)(63 136 72 127)
(1 63 35 44)(2 64 36 45)(3 65 19 46)(4 66 20 47)(5 67 21 48)(6 68 22 49)(7 69 23 50)(8 70 24 51)(9 71 25 52)(10 72 26 53)(11 55 27 54)(12 56 28 37)(13 57 29 38)(14 58 30 39)(15 59 31 40)(16 60 32 41)(17 61 33 42)(18 62 34 43)(73 144 104 126)(74 127 105 109)(75 128 106 110)(76 129 107 111)(77 130 108 112)(78 131 91 113)(79 132 92 114)(80 133 93 115)(81 134 94 116)(82 135 95 117)(83 136 96 118)(84 137 97 119)(85 138 98 120)(86 139 99 121)(87 140 100 122)(88 141 101 123)(89 142 102 124)(90 143 103 125)
G:=sub<Sym(144)| (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), (1,74,10,83)(2,73,11,82)(3,90,12,81)(4,89,13,80)(5,88,14,79)(6,87,15,78)(7,86,16,77)(8,85,17,76)(9,84,18,75)(19,103,28,94)(20,102,29,93)(21,101,30,92)(22,100,31,91)(23,99,32,108)(24,98,33,107)(25,97,34,106)(26,96,35,105)(27,95,36,104)(37,125,46,116)(38,124,47,115)(39,123,48,114)(40,122,49,113)(41,121,50,112)(42,120,51,111)(43,119,52,110)(44,118,53,109)(45,117,54,126)(55,144,64,135)(56,143,65,134)(57,142,66,133)(58,141,67,132)(59,140,68,131)(60,139,69,130)(61,138,70,129)(62,137,71,128)(63,136,72,127), (1,63,35,44)(2,64,36,45)(3,65,19,46)(4,66,20,47)(5,67,21,48)(6,68,22,49)(7,69,23,50)(8,70,24,51)(9,71,25,52)(10,72,26,53)(11,55,27,54)(12,56,28,37)(13,57,29,38)(14,58,30,39)(15,59,31,40)(16,60,32,41)(17,61,33,42)(18,62,34,43)(73,144,104,126)(74,127,105,109)(75,128,106,110)(76,129,107,111)(77,130,108,112)(78,131,91,113)(79,132,92,114)(80,133,93,115)(81,134,94,116)(82,135,95,117)(83,136,96,118)(84,137,97,119)(85,138,98,120)(86,139,99,121)(87,140,100,122)(88,141,101,123)(89,142,102,124)(90,143,103,125)>;
G:=Group( (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), (1,74,10,83)(2,73,11,82)(3,90,12,81)(4,89,13,80)(5,88,14,79)(6,87,15,78)(7,86,16,77)(8,85,17,76)(9,84,18,75)(19,103,28,94)(20,102,29,93)(21,101,30,92)(22,100,31,91)(23,99,32,108)(24,98,33,107)(25,97,34,106)(26,96,35,105)(27,95,36,104)(37,125,46,116)(38,124,47,115)(39,123,48,114)(40,122,49,113)(41,121,50,112)(42,120,51,111)(43,119,52,110)(44,118,53,109)(45,117,54,126)(55,144,64,135)(56,143,65,134)(57,142,66,133)(58,141,67,132)(59,140,68,131)(60,139,69,130)(61,138,70,129)(62,137,71,128)(63,136,72,127), (1,63,35,44)(2,64,36,45)(3,65,19,46)(4,66,20,47)(5,67,21,48)(6,68,22,49)(7,69,23,50)(8,70,24,51)(9,71,25,52)(10,72,26,53)(11,55,27,54)(12,56,28,37)(13,57,29,38)(14,58,30,39)(15,59,31,40)(16,60,32,41)(17,61,33,42)(18,62,34,43)(73,144,104,126)(74,127,105,109)(75,128,106,110)(76,129,107,111)(77,130,108,112)(78,131,91,113)(79,132,92,114)(80,133,93,115)(81,134,94,116)(82,135,95,117)(83,136,96,118)(84,137,97,119)(85,138,98,120)(86,139,99,121)(87,140,100,122)(88,141,101,123)(89,142,102,124)(90,143,103,125) );
G=PermutationGroup([[(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)], [(1,74,10,83),(2,73,11,82),(3,90,12,81),(4,89,13,80),(5,88,14,79),(6,87,15,78),(7,86,16,77),(8,85,17,76),(9,84,18,75),(19,103,28,94),(20,102,29,93),(21,101,30,92),(22,100,31,91),(23,99,32,108),(24,98,33,107),(25,97,34,106),(26,96,35,105),(27,95,36,104),(37,125,46,116),(38,124,47,115),(39,123,48,114),(40,122,49,113),(41,121,50,112),(42,120,51,111),(43,119,52,110),(44,118,53,109),(45,117,54,126),(55,144,64,135),(56,143,65,134),(57,142,66,133),(58,141,67,132),(59,140,68,131),(60,139,69,130),(61,138,70,129),(62,137,71,128),(63,136,72,127)], [(1,63,35,44),(2,64,36,45),(3,65,19,46),(4,66,20,47),(5,67,21,48),(6,68,22,49),(7,69,23,50),(8,70,24,51),(9,71,25,52),(10,72,26,53),(11,55,27,54),(12,56,28,37),(13,57,29,38),(14,58,30,39),(15,59,31,40),(16,60,32,41),(17,61,33,42),(18,62,34,43),(73,144,104,126),(74,127,105,109),(75,128,106,110),(76,129,107,111),(77,130,108,112),(78,131,91,113),(79,132,92,114),(80,133,93,115),(81,134,94,116),(82,135,95,117),(83,136,96,118),(84,137,97,119),(85,138,98,120),(86,139,99,121),(87,140,100,122),(88,141,101,123),(89,142,102,124),(90,143,103,125)]])
Dic9⋊C4 is a maximal subgroup of
C4×Dic18 C36.6Q8 C42⋊2D9 C42⋊3D9 C23.16D18 C22⋊2Dic18 C23.8D18 Dic9⋊4D4 C23.9D18 D18⋊D4 Dic9⋊3Q8 C36⋊Q8 Dic9.Q8 C36.3Q8 C4⋊C4×D9 D18.D4 D18⋊Q8 C4⋊C4⋊D9 C36.49D4 C4×C9⋊D4 C23.28D18 C23.23D18 Dic9⋊D4 Dic9⋊Q8 D18⋊3Q8 Dic27⋊C4 Dic9⋊Dic3 C18.Dic6 Dic9⋊C12 C6.Dic18
Dic9⋊C4 is a maximal quotient of
C36.Q8 C4.Dic18 Dic9⋊C8 C36.53D4 C18.C42 Dic27⋊C4 Dic9⋊Dic3 C18.Dic6 C6.Dic18
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | Q8 | D6 | D9 | Dic6 | C4×S3 | C3⋊D4 | D18 | Dic18 | C4×D9 | C9⋊D4 |
| kernel | Dic9⋊C4 | C2×Dic9 | C2×C36 | Dic9 | C2×C12 | C18 | C18 | C2×C6 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 2 | 3 | 6 | 6 | 6 |
Matrix representation of Dic9⋊C4 ►in GL4(𝔽37) generated by
| 26 | 0 | 0 | 0 |
| 24 | 10 | 0 | 0 |
| 0 | 0 | 11 | 31 |
| 0 | 0 | 6 | 17 |
| 14 | 3 | 0 | 0 |
| 9 | 23 | 0 | 0 |
| 0 | 0 | 7 | 30 |
| 0 | 0 | 23 | 30 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 32 | 27 |
| 0 | 0 | 10 | 5 |
G:=sub<GL(4,GF(37))| [26,24,0,0,0,10,0,0,0,0,11,6,0,0,31,17],[14,9,0,0,3,23,0,0,0,0,7,23,0,0,30,30],[6,0,0,0,0,6,0,0,0,0,32,10,0,0,27,5] >;
Dic9⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_9\rtimes C_4 % in TeX
G:=Group("Dic9:C4"); // GroupNames label
G:=SmallGroup(144,12);
// by ID
G=gap.SmallGroup(144,12);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,121,31,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c|a^18=c^4=1,b^2=a^9,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^9*b>;
// generators/relations
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