metabelian, supersoluble, monomial
Aliases: C12⋊3D12, C62.239C23, (C3×C12)⋊12D4, (C2×C12).34D6, C6.116(S3×D4), C6.54(C2×D12), C4⋊2(C12⋊S3), C3⋊2(C12⋊D4), C6.11D12⋊7C2, C32⋊19(C4⋊D4), (C6×C12).16C22, C6.51(Q8⋊3S3), C2.6(C12.26D6), (C3×C4⋊C4)⋊6S3, C4⋊C4⋊3(C3⋊S3), (C2×C3⋊S3)⋊12D4, C2.13(D4×C3⋊S3), (C2×C12⋊S3)⋊6C2, (C32×C4⋊C4)⋊15C2, C2.9(C2×C12⋊S3), (C3×C6).194(C2×D4), (C3×C6).161(C4○D4), (C2×C6).256(C22×S3), C22.50(C22×C3⋊S3), (C22×C3⋊S3).87C22, (C2×C3⋊Dic3).160C22, (C2×C4×C3⋊S3)⋊2C2, (C2×C4).12(C2×C3⋊S3), SmallGroup(288,752)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊3D12
G = < a,b,c | a12=b12=c2=1, bab-1=a7, cac=a-1, cbc=b-1 >
Subgroups: 1388 in 282 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C4⋊D4, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C6×C12, C6×C12, C22×C3⋊S3, C22×C3⋊S3, C12⋊D4, C6.11D12, C32×C4⋊C4, C2×C4×C3⋊S3, C2×C12⋊S3, C2×C12⋊S3, C12⋊3D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊S3, D12, C22×S3, C4⋊D4, C2×C3⋊S3, C2×D12, S3×D4, Q8⋊3S3, C12⋊S3, C22×C3⋊S3, C12⋊D4, C2×C12⋊S3, D4×C3⋊S3, C12.26D6, C12⋊3D12
►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 49 46 34 141 17 69 75 89 128 120 108)(2 56 47 29 142 24 70 82 90 123 109 103)(3 51 48 36 143 19 71 77 91 130 110 98)(4 58 37 31 144 14 72 84 92 125 111 105)(5 53 38 26 133 21 61 79 93 132 112 100)(6 60 39 33 134 16 62 74 94 127 113 107)(7 55 40 28 135 23 63 81 95 122 114 102)(8 50 41 35 136 18 64 76 96 129 115 97)(9 57 42 30 137 13 65 83 85 124 116 104)(10 52 43 25 138 20 66 78 86 131 117 99)(11 59 44 32 139 15 67 73 87 126 118 106)(12 54 45 27 140 22 68 80 88 121 119 101)
(1 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 48)(12 47)(13 132)(14 131)(15 130)(16 129)(17 128)(18 127)(19 126)(20 125)(21 124)(22 123)(23 122)(24 121)(25 105)(26 104)(27 103)(28 102)(29 101)(30 100)(31 99)(32 98)(33 97)(34 108)(35 107)(36 106)(50 60)(51 59)(52 58)(53 57)(54 56)(61 85)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 90)(69 89)(70 88)(71 87)(72 86)(73 77)(74 76)(78 84)(79 83)(80 82)(109 140)(110 139)(111 138)(112 137)(113 136)(114 135)(115 134)(116 133)(117 144)(118 143)(119 142)(120 141)
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), (1,49,46,34,141,17,69,75,89,128,120,108)(2,56,47,29,142,24,70,82,90,123,109,103)(3,51,48,36,143,19,71,77,91,130,110,98)(4,58,37,31,144,14,72,84,92,125,111,105)(5,53,38,26,133,21,61,79,93,132,112,100)(6,60,39,33,134,16,62,74,94,127,113,107)(7,55,40,28,135,23,63,81,95,122,114,102)(8,50,41,35,136,18,64,76,96,129,115,97)(9,57,42,30,137,13,65,83,85,124,116,104)(10,52,43,25,138,20,66,78,86,131,117,99)(11,59,44,32,139,15,67,73,87,126,118,106)(12,54,45,27,140,22,68,80,88,121,119,101), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,48)(12,47)(13,132)(14,131)(15,130)(16,129)(17,128)(18,127)(19,126)(20,125)(21,124)(22,123)(23,122)(24,121)(25,105)(26,104)(27,103)(28,102)(29,101)(30,100)(31,99)(32,98)(33,97)(34,108)(35,107)(36,106)(50,60)(51,59)(52,58)(53,57)(54,56)(61,85)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,87)(72,86)(73,77)(74,76)(78,84)(79,83)(80,82)(109,140)(110,139)(111,138)(112,137)(113,136)(114,135)(115,134)(116,133)(117,144)(118,143)(119,142)(120,141)>;
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), (1,49,46,34,141,17,69,75,89,128,120,108)(2,56,47,29,142,24,70,82,90,123,109,103)(3,51,48,36,143,19,71,77,91,130,110,98)(4,58,37,31,144,14,72,84,92,125,111,105)(5,53,38,26,133,21,61,79,93,132,112,100)(6,60,39,33,134,16,62,74,94,127,113,107)(7,55,40,28,135,23,63,81,95,122,114,102)(8,50,41,35,136,18,64,76,96,129,115,97)(9,57,42,30,137,13,65,83,85,124,116,104)(10,52,43,25,138,20,66,78,86,131,117,99)(11,59,44,32,139,15,67,73,87,126,118,106)(12,54,45,27,140,22,68,80,88,121,119,101), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,48)(12,47)(13,132)(14,131)(15,130)(16,129)(17,128)(18,127)(19,126)(20,125)(21,124)(22,123)(23,122)(24,121)(25,105)(26,104)(27,103)(28,102)(29,101)(30,100)(31,99)(32,98)(33,97)(34,108)(35,107)(36,106)(50,60)(51,59)(52,58)(53,57)(54,56)(61,85)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,87)(72,86)(73,77)(74,76)(78,84)(79,83)(80,82)(109,140)(110,139)(111,138)(112,137)(113,136)(114,135)(115,134)(116,133)(117,144)(118,143)(119,142)(120,141) );
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)], [(1,49,46,34,141,17,69,75,89,128,120,108),(2,56,47,29,142,24,70,82,90,123,109,103),(3,51,48,36,143,19,71,77,91,130,110,98),(4,58,37,31,144,14,72,84,92,125,111,105),(5,53,38,26,133,21,61,79,93,132,112,100),(6,60,39,33,134,16,62,74,94,127,113,107),(7,55,40,28,135,23,63,81,95,122,114,102),(8,50,41,35,136,18,64,76,96,129,115,97),(9,57,42,30,137,13,65,83,85,124,116,104),(10,52,43,25,138,20,66,78,86,131,117,99),(11,59,44,32,139,15,67,73,87,126,118,106),(12,54,45,27,140,22,68,80,88,121,119,101)], [(1,46),(2,45),(3,44),(4,43),(5,42),(6,41),(7,40),(8,39),(9,38),(10,37),(11,48),(12,47),(13,132),(14,131),(15,130),(16,129),(17,128),(18,127),(19,126),(20,125),(21,124),(22,123),(23,122),(24,121),(25,105),(26,104),(27,103),(28,102),(29,101),(30,100),(31,99),(32,98),(33,97),(34,108),(35,107),(36,106),(50,60),(51,59),(52,58),(53,57),(54,56),(61,85),(62,96),(63,95),(64,94),(65,93),(66,92),(67,91),(68,90),(69,89),(70,88),(71,87),(72,86),(73,77),(74,76),(78,84),(79,83),(80,82),(109,140),(110,139),(111,138),(112,137),(113,136),(114,135),(115,134),(116,133),(117,144),(118,143),(119,142),(120,141)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6L | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | C4○D4 | D12 | S3×D4 | Q8⋊3S3 |
| kernel | C12⋊3D12 | C6.11D12 | C32×C4⋊C4 | C2×C4×C3⋊S3 | C2×C12⋊S3 | C3×C4⋊C4 | C3×C12 | C2×C3⋊S3 | C2×C12 | C3×C6 | C12 | C6 | C6 |
| # reps | 1 | 2 | 1 | 1 | 3 | 4 | 2 | 2 | 12 | 2 | 16 | 4 | 4 |
Matrix representation of C12⋊3D12 ►in GL8(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 2 |
| 12 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 12 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,11,3,0,0,0,0,0,0,7,2],[12,5,0,0,0,0,0,0,10,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,1],[12,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,12] >;
C12⋊3D12 in GAP, Magma, Sage, TeX
C_{12}\rtimes_3D_{12} % in TeX
G:=Group("C12:3D12"); // GroupNames label
G:=SmallGroup(288,752);
// by ID
G=gap.SmallGroup(288,752);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,58,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=c^2=1,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations