metabelian, supersoluble, monomial
Aliases: Dic9⋊C12, C62.23D6, C9⋊C12⋊C4, Dic9⋊C4⋊C3, (C2×C36).8C6, (C6×C12).2S3, C18.6(C3×D4), C18.1(C3×Q8), C6.18(S3×C12), C18.3(C2×C12), C6.7(C3×Dic6), (C3×C6).7Dic6, (C2×Dic9).1C6, C2.1(Dic9⋊C6), C32.(Dic3⋊C4), C2.1(C36.C6), 3- 1+2⋊1(C4⋊C4), (C2×3- 1+2).1Q8, (C2×3- 1+2).6D4, (C22×3- 1+2).3C22, C9⋊1(C3×C4⋊C4), C2.4(C4×C9⋊C6), (C2×C9⋊C12).1C2, (C2×C4).1(C9⋊C6), (C2×C18).3(C2×C6), (C2×C12).7(C3×S3), (C3×C6).18(C4×S3), (C2×C6).43(S3×C6), C6.16(C3×C3⋊D4), C22.4(C2×C9⋊C6), C3.3(C3×Dic3⋊C4), (C3×C6).21(C3⋊D4), (C2×C4×3- 1+2).8C2, (C2×3- 1+2).3(C2×C4), SmallGroup(432,145)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic9⋊C12
G = < a,b,c | a18=c12=1, b2=a9, bab-1=a-1, cac-1=a13, cbc-1=a9b >
Subgroups: 262 in 86 conjugacy classes, 40 normal (36 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C2×C4, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C18, C18, C3×C6, C2×Dic3, C2×C12, C2×C12, 3- 1+2, Dic9, Dic9, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, Dic3⋊C4, C3×C4⋊C4, C2×3- 1+2, C2×Dic9, C2×C36, C2×C36, C6×Dic3, C6×C12, C9⋊C12, C9⋊C12, C4×3- 1+2, C22×3- 1+2, Dic9⋊C4, C3×Dic3⋊C4, C2×C9⋊C12, C2×C4×3- 1+2, Dic9⋊C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C12, D6, C2×C6, C4⋊C4, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, S3×C6, Dic3⋊C4, C3×C4⋊C4, C9⋊C6, C3×Dic6, S3×C12, C3×C3⋊D4, C2×C9⋊C6, C3×Dic3⋊C4, C36.C6, C4×C9⋊C6, Dic9⋊C6, Dic9⋊C12
►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 82 10 73)(2 81 11 90)(3 80 12 89)(4 79 13 88)(5 78 14 87)(6 77 15 86)(7 76 16 85)(8 75 17 84)(9 74 18 83)(19 92 28 101)(20 91 29 100)(21 108 30 99)(22 107 31 98)(23 106 32 97)(24 105 33 96)(25 104 34 95)(26 103 35 94)(27 102 36 93)(37 120 46 111)(38 119 47 110)(39 118 48 109)(40 117 49 126)(41 116 50 125)(42 115 51 124)(43 114 52 123)(44 113 53 122)(45 112 54 121)(55 137 64 128)(56 136 65 127)(57 135 66 144)(58 134 67 143)(59 133 68 142)(60 132 69 141)(61 131 70 140)(62 130 71 139)(63 129 72 138)
(1 69 19 44)(2 58 32 45 8 64 20 51 14 70 26 39)(3 65 27 46 15 59 21 40 9 71 33 52)(4 72 22 47)(5 61 35 48 11 67 23 54 17 55 29 42)(6 68 30 49 18 62 24 43 12 56 36 37)(7 57 25 50)(10 60 28 53)(13 63 31 38)(16 66 34 41)(73 141 101 113)(74 130 96 114 80 136 102 120 86 142 108 126)(75 137 91 115 87 131 103 109 81 143 97 121)(76 144 104 116)(77 133 99 117 83 139 105 123 89 127 93 111)(78 140 94 118 90 134 106 112 84 128 100 124)(79 129 107 119)(82 132 92 122)(85 135 95 125)(88 138 98 110)
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), (1,82,10,73)(2,81,11,90)(3,80,12,89)(4,79,13,88)(5,78,14,87)(6,77,15,86)(7,76,16,85)(8,75,17,84)(9,74,18,83)(19,92,28,101)(20,91,29,100)(21,108,30,99)(22,107,31,98)(23,106,32,97)(24,105,33,96)(25,104,34,95)(26,103,35,94)(27,102,36,93)(37,120,46,111)(38,119,47,110)(39,118,48,109)(40,117,49,126)(41,116,50,125)(42,115,51,124)(43,114,52,123)(44,113,53,122)(45,112,54,121)(55,137,64,128)(56,136,65,127)(57,135,66,144)(58,134,67,143)(59,133,68,142)(60,132,69,141)(61,131,70,140)(62,130,71,139)(63,129,72,138), (1,69,19,44)(2,58,32,45,8,64,20,51,14,70,26,39)(3,65,27,46,15,59,21,40,9,71,33,52)(4,72,22,47)(5,61,35,48,11,67,23,54,17,55,29,42)(6,68,30,49,18,62,24,43,12,56,36,37)(7,57,25,50)(10,60,28,53)(13,63,31,38)(16,66,34,41)(73,141,101,113)(74,130,96,114,80,136,102,120,86,142,108,126)(75,137,91,115,87,131,103,109,81,143,97,121)(76,144,104,116)(77,133,99,117,83,139,105,123,89,127,93,111)(78,140,94,118,90,134,106,112,84,128,100,124)(79,129,107,119)(82,132,92,122)(85,135,95,125)(88,138,98,110)>;
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), (1,82,10,73)(2,81,11,90)(3,80,12,89)(4,79,13,88)(5,78,14,87)(6,77,15,86)(7,76,16,85)(8,75,17,84)(9,74,18,83)(19,92,28,101)(20,91,29,100)(21,108,30,99)(22,107,31,98)(23,106,32,97)(24,105,33,96)(25,104,34,95)(26,103,35,94)(27,102,36,93)(37,120,46,111)(38,119,47,110)(39,118,48,109)(40,117,49,126)(41,116,50,125)(42,115,51,124)(43,114,52,123)(44,113,53,122)(45,112,54,121)(55,137,64,128)(56,136,65,127)(57,135,66,144)(58,134,67,143)(59,133,68,142)(60,132,69,141)(61,131,70,140)(62,130,71,139)(63,129,72,138), (1,69,19,44)(2,58,32,45,8,64,20,51,14,70,26,39)(3,65,27,46,15,59,21,40,9,71,33,52)(4,72,22,47)(5,61,35,48,11,67,23,54,17,55,29,42)(6,68,30,49,18,62,24,43,12,56,36,37)(7,57,25,50)(10,60,28,53)(13,63,31,38)(16,66,34,41)(73,141,101,113)(74,130,96,114,80,136,102,120,86,142,108,126)(75,137,91,115,87,131,103,109,81,143,97,121)(76,144,104,116)(77,133,99,117,83,139,105,123,89,127,93,111)(78,140,94,118,90,134,106,112,84,128,100,124)(79,129,107,119)(82,132,92,122)(85,135,95,125)(88,138,98,110) );
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)], [(1,82,10,73),(2,81,11,90),(3,80,12,89),(4,79,13,88),(5,78,14,87),(6,77,15,86),(7,76,16,85),(8,75,17,84),(9,74,18,83),(19,92,28,101),(20,91,29,100),(21,108,30,99),(22,107,31,98),(23,106,32,97),(24,105,33,96),(25,104,34,95),(26,103,35,94),(27,102,36,93),(37,120,46,111),(38,119,47,110),(39,118,48,109),(40,117,49,126),(41,116,50,125),(42,115,51,124),(43,114,52,123),(44,113,53,122),(45,112,54,121),(55,137,64,128),(56,136,65,127),(57,135,66,144),(58,134,67,143),(59,133,68,142),(60,132,69,141),(61,131,70,140),(62,130,71,139),(63,129,72,138)], [(1,69,19,44),(2,58,32,45,8,64,20,51,14,70,26,39),(3,65,27,46,15,59,21,40,9,71,33,52),(4,72,22,47),(5,61,35,48,11,67,23,54,17,55,29,42),(6,68,30,49,18,62,24,43,12,56,36,37),(7,57,25,50),(10,60,28,53),(13,63,31,38),(16,66,34,41),(73,141,101,113),(74,130,96,114,80,136,102,120,86,142,108,126),(75,137,91,115,87,131,103,109,81,143,97,121),(76,144,104,116),(77,133,99,117,83,139,105,123,89,127,93,111),(78,140,94,118,90,134,106,112,84,128,100,124),(79,129,107,119),(82,132,92,122),(85,135,95,125),(88,138,98,110)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | ··· | 6I | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12P | 18A | ··· | 18I | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 18 | ··· | 18 | 6 | ··· | 6 | 6 | ··· | 6 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | - | + | - | + | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | D4 | Q8 | D6 | C3×S3 | C3×D4 | C3×Q8 | Dic6 | C4×S3 | C3⋊D4 | S3×C6 | C3×Dic6 | S3×C12 | C3×C3⋊D4 | C9⋊C6 | C2×C9⋊C6 | C36.C6 | C4×C9⋊C6 | Dic9⋊C6 |
| kernel | Dic9⋊C12 | C2×C9⋊C12 | C2×C4×3- 1+2 | Dic9⋊C4 | C9⋊C12 | C2×Dic9 | C2×C36 | Dic9 | C6×C12 | C2×3- 1+2 | C2×3- 1+2 | C62 | C2×C12 | C18 | C18 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of Dic9⋊C12 ►in GL8(𝔽37)
| 36 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 36 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 | 28 |
| 0 | 0 | 0 | 0 | 0 | 0 | 29 | 1 |
| 0 | 0 | 0 | 0 | 36 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 1 | 0 | 0 |
| 29 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 29 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 14 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 30 | 7 |
G:=sub<GL(8,GF(37))| [36,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,1,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,29,9,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,36,29,0,0,0,0,0,0,28,1,0,0,0,0,36,29,0,0,0,0,0,0,28,1,0,0],[29,0,0,0,0,0,0,0,0,29,0,0,0,0,0,0,0,0,30,14,0,0,0,0,0,0,23,7,0,0,0,0,0,0,0,0,30,30,0,0,0,0,0,0,7,23,0,0,0,0,0,0,0,0,14,30,0,0,0,0,0,0,7,7] >;
Dic9⋊C12 in GAP, Magma, Sage, TeX
{\rm Dic}_9\rtimes C_{12} % in TeX
G:=Group("Dic9:C12"); // GroupNames label
G:=SmallGroup(432,145);
// by ID
G=gap.SmallGroup(432,145);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,365,92,10085,2035,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^18=c^12=1,b^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^9*b>;
// generators/relations