metabelian, supersoluble, monomial
Aliases: C12⋊4D12, C122⋊5C2, C62.218C23, (C4×C12)⋊6S3, (C3×C12)⋊17D4, C42⋊6(C3⋊S3), C6.50(C2×D12), C4⋊1(C12⋊S3), C3⋊1(C4⋊D12), (C2×C12).383D6, C32⋊6(C4⋊1D4), (C6×C12).299C22, (C2×C12⋊S3)⋊3C2, C2.5(C2×C12⋊S3), (C3×C6).190(C2×D4), (C2×C6).235(C22×S3), C22.36(C22×C3⋊S3), (C22×C3⋊S3).37C22, (C2×C4).77(C2×C3⋊S3), SmallGroup(288,731)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊4D12
G = < a,b,c | a12=b12=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 1740 in 324 conjugacy classes, 101 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C32, C12, D6, C2×C6, C42, C2×D4, C3⋊S3, C3×C6, D12, C2×C12, C22×S3, C4⋊1D4, C3×C12, C2×C3⋊S3, C62, C4×C12, C2×D12, C12⋊S3, C6×C12, C22×C3⋊S3, C4⋊D12, C122, C2×C12⋊S3, C12⋊4D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, D12, C22×S3, C4⋊1D4, C2×C3⋊S3, C2×D12, C12⋊S3, C22×C3⋊S3, C4⋊D12, C2×C12⋊S3, C12⋊4D12
►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 22 143 27 93 112 69 84 130 101 44 50)(2 23 144 28 94 113 70 73 131 102 45 51)(3 24 133 29 95 114 71 74 132 103 46 52)(4 13 134 30 96 115 72 75 121 104 47 53)(5 14 135 31 85 116 61 76 122 105 48 54)(6 15 136 32 86 117 62 77 123 106 37 55)(7 16 137 33 87 118 63 78 124 107 38 56)(8 17 138 34 88 119 64 79 125 108 39 57)(9 18 139 35 89 120 65 80 126 97 40 58)(10 19 140 36 90 109 66 81 127 98 41 59)(11 20 141 25 91 110 67 82 128 99 42 60)(12 21 142 26 92 111 68 83 129 100 43 49)
(1 53)(2 52)(3 51)(4 50)(5 49)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 48)(22 47)(23 46)(24 45)(25 123)(26 122)(27 121)(28 132)(29 131)(30 130)(31 129)(32 128)(33 127)(34 126)(35 125)(36 124)(61 111)(62 110)(63 109)(64 120)(65 119)(66 118)(67 117)(68 116)(69 115)(70 114)(71 113)(72 112)(73 95)(74 94)(75 93)(76 92)(77 91)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(84 96)(97 138)(98 137)(99 136)(100 135)(101 134)(102 133)(103 144)(104 143)(105 142)(106 141)(107 140)(108 139)
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), (1,22,143,27,93,112,69,84,130,101,44,50)(2,23,144,28,94,113,70,73,131,102,45,51)(3,24,133,29,95,114,71,74,132,103,46,52)(4,13,134,30,96,115,72,75,121,104,47,53)(5,14,135,31,85,116,61,76,122,105,48,54)(6,15,136,32,86,117,62,77,123,106,37,55)(7,16,137,33,87,118,63,78,124,107,38,56)(8,17,138,34,88,119,64,79,125,108,39,57)(9,18,139,35,89,120,65,80,126,97,40,58)(10,19,140,36,90,109,66,81,127,98,41,59)(11,20,141,25,91,110,67,82,128,99,42,60)(12,21,142,26,92,111,68,83,129,100,43,49), (1,53)(2,52)(3,51)(4,50)(5,49)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,48)(22,47)(23,46)(24,45)(25,123)(26,122)(27,121)(28,132)(29,131)(30,130)(31,129)(32,128)(33,127)(34,126)(35,125)(36,124)(61,111)(62,110)(63,109)(64,120)(65,119)(66,118)(67,117)(68,116)(69,115)(70,114)(71,113)(72,112)(73,95)(74,94)(75,93)(76,92)(77,91)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)(84,96)(97,138)(98,137)(99,136)(100,135)(101,134)(102,133)(103,144)(104,143)(105,142)(106,141)(107,140)(108,139)>;
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), (1,22,143,27,93,112,69,84,130,101,44,50)(2,23,144,28,94,113,70,73,131,102,45,51)(3,24,133,29,95,114,71,74,132,103,46,52)(4,13,134,30,96,115,72,75,121,104,47,53)(5,14,135,31,85,116,61,76,122,105,48,54)(6,15,136,32,86,117,62,77,123,106,37,55)(7,16,137,33,87,118,63,78,124,107,38,56)(8,17,138,34,88,119,64,79,125,108,39,57)(9,18,139,35,89,120,65,80,126,97,40,58)(10,19,140,36,90,109,66,81,127,98,41,59)(11,20,141,25,91,110,67,82,128,99,42,60)(12,21,142,26,92,111,68,83,129,100,43,49), (1,53)(2,52)(3,51)(4,50)(5,49)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,48)(22,47)(23,46)(24,45)(25,123)(26,122)(27,121)(28,132)(29,131)(30,130)(31,129)(32,128)(33,127)(34,126)(35,125)(36,124)(61,111)(62,110)(63,109)(64,120)(65,119)(66,118)(67,117)(68,116)(69,115)(70,114)(71,113)(72,112)(73,95)(74,94)(75,93)(76,92)(77,91)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)(84,96)(97,138)(98,137)(99,136)(100,135)(101,134)(102,133)(103,144)(104,143)(105,142)(106,141)(107,140)(108,139) );
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)], [(1,22,143,27,93,112,69,84,130,101,44,50),(2,23,144,28,94,113,70,73,131,102,45,51),(3,24,133,29,95,114,71,74,132,103,46,52),(4,13,134,30,96,115,72,75,121,104,47,53),(5,14,135,31,85,116,61,76,122,105,48,54),(6,15,136,32,86,117,62,77,123,106,37,55),(7,16,137,33,87,118,63,78,124,107,38,56),(8,17,138,34,88,119,64,79,125,108,39,57),(9,18,139,35,89,120,65,80,126,97,40,58),(10,19,140,36,90,109,66,81,127,98,41,59),(11,20,141,25,91,110,67,82,128,99,42,60),(12,21,142,26,92,111,68,83,129,100,43,49)], [(1,53),(2,52),(3,51),(4,50),(5,49),(6,60),(7,59),(8,58),(9,57),(10,56),(11,55),(12,54),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,48),(22,47),(23,46),(24,45),(25,123),(26,122),(27,121),(28,132),(29,131),(30,130),(31,129),(32,128),(33,127),(34,126),(35,125),(36,124),(61,111),(62,110),(63,109),(64,120),(65,119),(66,118),(67,117),(68,116),(69,115),(70,114),(71,113),(72,112),(73,95),(74,94),(75,93),(76,92),(77,91),(78,90),(79,89),(80,88),(81,87),(82,86),(83,85),(84,96),(97,138),(98,137),(99,136),(100,135),(101,134),(102,133),(103,144),(104,143),(105,142),(106,141),(107,140),(108,139)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | ··· | 4F | 6A | ··· | 6L | 12A | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D4 | D6 | D12 |
| kernel | C12⋊4D12 | C122 | C2×C12⋊S3 | C4×C12 | C3×C12 | C2×C12 | C12 |
| # reps | 1 | 1 | 6 | 4 | 6 | 12 | 48 |
Matrix representation of C12⋊4D12 ►in GL4(𝔽13) generated by
| 0 | 12 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 10 | 6 |
| 0 | 0 | 7 | 3 |
| 3 | 6 | 0 | 0 |
| 7 | 10 | 0 | 0 |
| 0 | 0 | 6 | 10 |
| 0 | 0 | 3 | 3 |
| 10 | 7 | 0 | 0 |
| 10 | 3 | 0 | 0 |
| 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [0,1,0,0,12,1,0,0,0,0,10,7,0,0,6,3],[3,7,0,0,6,10,0,0,0,0,6,3,0,0,10,3],[10,10,0,0,7,3,0,0,0,0,12,0,0,0,1,1] >;
C12⋊4D12 in GAP, Magma, Sage, TeX
C_{12}\rtimes_4D_{12} % in TeX
G:=Group("C12:4D12"); // GroupNames label
G:=SmallGroup(288,731);
// by ID
G=gap.SmallGroup(288,731);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,58,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations