direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C19⋊C12, C38⋊C12, Dic19⋊3C6, (C2×C38).C6, C19⋊2(C2×C12), (C2×Dic19)⋊C3, C38.4(C2×C6), C22.(C19⋊C6), (C2×C19⋊C3)⋊C4, C19⋊C3⋊2(C2×C4), C2.2(C2×C19⋊C6), (C22×C19⋊C3).C2, (C2×C19⋊C3).4C22, SmallGroup(456,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C2×C19⋊C12 |
Generators and relations for C2×C19⋊C12
G = < a,b,c | a2=b19=c12=1, ab=ba, ac=ca, cbc-1=b8 >
Smallest permutation representation of C2×C19⋊C12
►On 152 points
Generators in S152
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(77 115)(78 116)(79 117)(80 118)(81 119)(82 120)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(89 127)(90 128)(91 129)(92 130)(93 131)(94 132)(95 133)(96 143)(97 144)(98 145)(99 146)(100 147)(101 148)(102 149)(103 150)(104 151)(105 152)(106 134)(107 135)(108 136)(109 137)(110 138)(111 139)(112 140)(113 141)(114 142)
(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)(134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152)
(1 99 20 86)(2 111 31 85 8 107 21 79 12 98 27 94)(3 104 23 84 15 96 22 91 4 97 34 83)(5 109 26 82 10 112 24 77 7 114 29 80)(6 102 37 81 17 101 25 89 18 113 36 88)(9 100 32 78 19 106 28 87 13 110 38 93)(11 105 35 95 14 103 30 92 16 108 33 90)(39 146 58 124)(40 139 69 123 46 135 59 117 50 145 65 132)(41 151 61 122 53 143 60 129 42 144 72 121)(43 137 64 120 48 140 62 115 45 142 67 118)(44 149 75 119 55 148 63 127 56 141 74 126)(47 147 70 116 57 134 66 125 51 138 76 131)(49 152 73 133 52 150 68 130 54 136 71 128)
G:=sub<Sym(152)| (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(77,115)(78,116)(79,117)(80,118)(81,119)(82,120)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,143)(97,144)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,134)(107,135)(108,136)(109,137)(110,138)(111,139)(112,140)(113,141)(114,142), (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)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (1,99,20,86)(2,111,31,85,8,107,21,79,12,98,27,94)(3,104,23,84,15,96,22,91,4,97,34,83)(5,109,26,82,10,112,24,77,7,114,29,80)(6,102,37,81,17,101,25,89,18,113,36,88)(9,100,32,78,19,106,28,87,13,110,38,93)(11,105,35,95,14,103,30,92,16,108,33,90)(39,146,58,124)(40,139,69,123,46,135,59,117,50,145,65,132)(41,151,61,122,53,143,60,129,42,144,72,121)(43,137,64,120,48,140,62,115,45,142,67,118)(44,149,75,119,55,148,63,127,56,141,74,126)(47,147,70,116,57,134,66,125,51,138,76,131)(49,152,73,133,52,150,68,130,54,136,71,128)>;
G:=Group( (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(77,115)(78,116)(79,117)(80,118)(81,119)(82,120)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,143)(97,144)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,134)(107,135)(108,136)(109,137)(110,138)(111,139)(112,140)(113,141)(114,142), (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)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (1,99,20,86)(2,111,31,85,8,107,21,79,12,98,27,94)(3,104,23,84,15,96,22,91,4,97,34,83)(5,109,26,82,10,112,24,77,7,114,29,80)(6,102,37,81,17,101,25,89,18,113,36,88)(9,100,32,78,19,106,28,87,13,110,38,93)(11,105,35,95,14,103,30,92,16,108,33,90)(39,146,58,124)(40,139,69,123,46,135,59,117,50,145,65,132)(41,151,61,122,53,143,60,129,42,144,72,121)(43,137,64,120,48,140,62,115,45,142,67,118)(44,149,75,119,55,148,63,127,56,141,74,126)(47,147,70,116,57,134,66,125,51,138,76,131)(49,152,73,133,52,150,68,130,54,136,71,128) );
G=PermutationGroup([[(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(31,69),(32,70),(33,71),(34,72),(35,73),(36,74),(37,75),(38,76),(77,115),(78,116),(79,117),(80,118),(81,119),(82,120),(83,121),(84,122),(85,123),(86,124),(87,125),(88,126),(89,127),(90,128),(91,129),(92,130),(93,131),(94,132),(95,133),(96,143),(97,144),(98,145),(99,146),(100,147),(101,148),(102,149),(103,150),(104,151),(105,152),(106,134),(107,135),(108,136),(109,137),(110,138),(111,139),(112,140),(113,141),(114,142)], [(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),(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)], [(1,99,20,86),(2,111,31,85,8,107,21,79,12,98,27,94),(3,104,23,84,15,96,22,91,4,97,34,83),(5,109,26,82,10,112,24,77,7,114,29,80),(6,102,37,81,17,101,25,89,18,113,36,88),(9,100,32,78,19,106,28,87,13,110,38,93),(11,105,35,95,14,103,30,92,16,108,33,90),(39,146,58,124),(40,139,69,123,46,135,59,117,50,145,65,132),(41,151,61,122,53,143,60,129,42,144,72,121),(43,137,64,120,48,140,62,115,45,142,67,118),(44,149,75,119,55,148,63,127,56,141,74,126),(47,147,70,116,57,134,66,125,51,138,76,131),(49,152,73,133,52,150,68,130,54,136,71,128)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 12A | ··· | 12H | 19A | 19B | 19C | 38A | ··· | 38I |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 19 | 19 | 19 | 38 | ··· | 38 |
| size | 1 | 1 | 1 | 1 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | ··· | 19 | 19 | ··· | 19 | 6 | 6 | 6 | 6 | ··· | 6 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | C19⋊C6 | C19⋊C12 | C2×C19⋊C6 |
| kernel | C2×C19⋊C12 | C19⋊C12 | C22×C19⋊C3 | C2×Dic19 | C2×C19⋊C3 | Dic19 | C2×C38 | C38 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 3 | 6 | 3 |
Matrix representation of C2×C19⋊C12 ►in GL7(𝔽229)
| 228 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 228 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 228 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 228 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 228 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 228 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 101 | 106 | 83 | 106 | 101 | 228 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 203 | 49 | 159 | 12 | 212 | 139 |
| 0 | 137 | 15 | 191 | 142 | 128 | 9 |
| 0 | 165 | 46 | 187 | 198 | 44 | 211 |
| 0 | 90 | 17 | 217 | 70 | 180 | 81 |
| 0 | 5 | 58 | 51 | 113 | 36 | 104 |
| 0 | 176 | 176 | 166 | 175 | 139 | 176 |
G:=sub<GL(7,GF(229))| [228,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228],[1,0,0,0,0,0,0,0,101,1,0,0,0,0,0,106,0,1,0,0,0,0,83,0,0,1,0,0,0,106,0,0,0,1,0,0,101,0,0,0,0,1,0,228,0,0,0,0,0],[18,0,0,0,0,0,0,0,203,137,165,90,5,176,0,49,15,46,17,58,176,0,159,191,187,217,51,166,0,12,142,198,70,113,175,0,212,128,44,180,36,139,0,139,9,211,81,104,176] >;
C2×C19⋊C12 in GAP, Magma, Sage, TeX
C_2\times C_{19}\rtimes C_{12} % in TeX
G:=Group("C2xC19:C12"); // GroupNames label
G:=SmallGroup(456,10);
// by ID
G=gap.SmallGroup(456,10);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-19,60,10804,1064]);
// Polycyclic
G:=Group<a,b,c|a^2=b^19=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^8>;
// generators/relations
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