direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×C26.C6, C39⋊4C12, C78.4C6, Dic13⋊C32, C26.(C3×C6), C13⋊C3⋊2C12, C13⋊2(C3×C12), C6.4(C13⋊C6), (C3×Dic13)⋊C3, C2.(C3×C13⋊C6), (C3×C13⋊C3)⋊4C4, (C2×C13⋊C3).C6, (C6×C13⋊C3).2C2, SmallGroup(468,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C3×C26.C6 |
Generators and relations for C3×C26.C6
G = < a,b,c | a3=b26=1, c6=b13, ab=ba, ac=ca, cbc-1=b23 >
Smallest permutation representation of C3×C26.C6
►On 156 points
Generators in S156
(1 53 45)(2 54 46)(3 55 47)(4 56 48)(5 57 49)(6 58 50)(7 59 51)(8 60 52)(9 61 27)(10 62 28)(11 63 29)(12 64 30)(13 65 31)(14 66 32)(15 67 33)(16 68 34)(17 69 35)(18 70 36)(19 71 37)(20 72 38)(21 73 39)(22 74 40)(23 75 41)(24 76 42)(25 77 43)(26 78 44)(79 143 126)(80 144 127)(81 145 128)(82 146 129)(83 147 130)(84 148 105)(85 149 106)(86 150 107)(87 151 108)(88 152 109)(89 153 110)(90 154 111)(91 155 112)(92 156 113)(93 131 114)(94 132 115)(95 133 116)(96 134 117)(97 135 118)(98 136 119)(99 137 120)(100 138 121)(101 139 122)(102 140 123)(103 141 124)(104 142 125)
(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)(131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156)
(1 122 66 88 45 139 14 109 53 101 32 152)(2 113 69 87 28 136 15 126 56 100 41 149)(3 130 72 86 37 133 16 117 59 99 50 146)(4 121 75 85 46 156 17 108 62 98 33 143)(5 112 78 84 29 153 18 125 65 97 42 140)(6 129 55 83 38 150 19 116 68 96 51 137)(7 120 58 82 47 147 20 107 71 95 34 134)(8 111 61 81 30 144 21 124 74 94 43 131)(9 128 64 80 39 141 22 115 77 93 52 154)(10 119 67 79 48 138 23 106 54 92 35 151)(11 110 70 104 31 135 24 123 57 91 44 148)(12 127 73 103 40 132 25 114 60 90 27 145)(13 118 76 102 49 155 26 105 63 89 36 142)
G:=sub<Sym(156)| (1,53,45)(2,54,46)(3,55,47)(4,56,48)(5,57,49)(6,58,50)(7,59,51)(8,60,52)(9,61,27)(10,62,28)(11,63,29)(12,64,30)(13,65,31)(14,66,32)(15,67,33)(16,68,34)(17,69,35)(18,70,36)(19,71,37)(20,72,38)(21,73,39)(22,74,40)(23,75,41)(24,76,42)(25,77,43)(26,78,44)(79,143,126)(80,144,127)(81,145,128)(82,146,129)(83,147,130)(84,148,105)(85,149,106)(86,150,107)(87,151,108)(88,152,109)(89,153,110)(90,154,111)(91,155,112)(92,156,113)(93,131,114)(94,132,115)(95,133,116)(96,134,117)(97,135,118)(98,136,119)(99,137,120)(100,138,121)(101,139,122)(102,140,123)(103,141,124)(104,142,125), (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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,122,66,88,45,139,14,109,53,101,32,152)(2,113,69,87,28,136,15,126,56,100,41,149)(3,130,72,86,37,133,16,117,59,99,50,146)(4,121,75,85,46,156,17,108,62,98,33,143)(5,112,78,84,29,153,18,125,65,97,42,140)(6,129,55,83,38,150,19,116,68,96,51,137)(7,120,58,82,47,147,20,107,71,95,34,134)(8,111,61,81,30,144,21,124,74,94,43,131)(9,128,64,80,39,141,22,115,77,93,52,154)(10,119,67,79,48,138,23,106,54,92,35,151)(11,110,70,104,31,135,24,123,57,91,44,148)(12,127,73,103,40,132,25,114,60,90,27,145)(13,118,76,102,49,155,26,105,63,89,36,142)>;
G:=Group( (1,53,45)(2,54,46)(3,55,47)(4,56,48)(5,57,49)(6,58,50)(7,59,51)(8,60,52)(9,61,27)(10,62,28)(11,63,29)(12,64,30)(13,65,31)(14,66,32)(15,67,33)(16,68,34)(17,69,35)(18,70,36)(19,71,37)(20,72,38)(21,73,39)(22,74,40)(23,75,41)(24,76,42)(25,77,43)(26,78,44)(79,143,126)(80,144,127)(81,145,128)(82,146,129)(83,147,130)(84,148,105)(85,149,106)(86,150,107)(87,151,108)(88,152,109)(89,153,110)(90,154,111)(91,155,112)(92,156,113)(93,131,114)(94,132,115)(95,133,116)(96,134,117)(97,135,118)(98,136,119)(99,137,120)(100,138,121)(101,139,122)(102,140,123)(103,141,124)(104,142,125), (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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,122,66,88,45,139,14,109,53,101,32,152)(2,113,69,87,28,136,15,126,56,100,41,149)(3,130,72,86,37,133,16,117,59,99,50,146)(4,121,75,85,46,156,17,108,62,98,33,143)(5,112,78,84,29,153,18,125,65,97,42,140)(6,129,55,83,38,150,19,116,68,96,51,137)(7,120,58,82,47,147,20,107,71,95,34,134)(8,111,61,81,30,144,21,124,74,94,43,131)(9,128,64,80,39,141,22,115,77,93,52,154)(10,119,67,79,48,138,23,106,54,92,35,151)(11,110,70,104,31,135,24,123,57,91,44,148)(12,127,73,103,40,132,25,114,60,90,27,145)(13,118,76,102,49,155,26,105,63,89,36,142) );
G=PermutationGroup([[(1,53,45),(2,54,46),(3,55,47),(4,56,48),(5,57,49),(6,58,50),(7,59,51),(8,60,52),(9,61,27),(10,62,28),(11,63,29),(12,64,30),(13,65,31),(14,66,32),(15,67,33),(16,68,34),(17,69,35),(18,70,36),(19,71,37),(20,72,38),(21,73,39),(22,74,40),(23,75,41),(24,76,42),(25,77,43),(26,78,44),(79,143,126),(80,144,127),(81,145,128),(82,146,129),(83,147,130),(84,148,105),(85,149,106),(86,150,107),(87,151,108),(88,152,109),(89,153,110),(90,154,111),(91,155,112),(92,156,113),(93,131,114),(94,132,115),(95,133,116),(96,134,117),(97,135,118),(98,136,119),(99,137,120),(100,138,121),(101,139,122),(102,140,123),(103,141,124),(104,142,125)], [(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),(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)], [(1,122,66,88,45,139,14,109,53,101,32,152),(2,113,69,87,28,136,15,126,56,100,41,149),(3,130,72,86,37,133,16,117,59,99,50,146),(4,121,75,85,46,156,17,108,62,98,33,143),(5,112,78,84,29,153,18,125,65,97,42,140),(6,129,55,83,38,150,19,116,68,96,51,137),(7,120,58,82,47,147,20,107,71,95,34,134),(8,111,61,81,30,144,21,124,74,94,43,131),(9,128,64,80,39,141,22,115,77,93,52,154),(10,119,67,79,48,138,23,106,54,92,35,151),(11,110,70,104,31,135,24,123,57,91,44,148),(12,127,73,103,40,132,25,114,60,90,27,145),(13,118,76,102,49,155,26,105,63,89,36,142)]])
48 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 6A | 6B | 6C | ··· | 6H | 12A | ··· | 12P | 13A | 13B | 26A | 26B | 39A | 39B | 39C | 39D | 78A | 78B | 78C | 78D |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 13 | 13 | 26 | 26 | 39 | 39 | 39 | 39 | 78 | 78 | 78 | 78 |
| size | 1 | 1 | 1 | 1 | 13 | ··· | 13 | 13 | 13 | 1 | 1 | 13 | ··· | 13 | 13 | ··· | 13 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 |
| type | + | + | + | - | |||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C12 | C12 | C13⋊C6 | C26.C6 | C3×C13⋊C6 | C3×C26.C6 |
| kernel | C3×C26.C6 | C6×C13⋊C3 | C26.C6 | C3×Dic13 | C3×C13⋊C3 | C2×C13⋊C3 | C78 | C13⋊C3 | C39 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 6 | 2 | 2 | 6 | 2 | 12 | 4 | 2 | 2 | 4 | 4 |
Matrix representation of C3×C26.C6 ►in GL7(𝔽157)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 144 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 144 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 144 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 144 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 144 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 144 |
| 156 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 90 | 66 | 92 | 65 | 91 | 67 |
| 0 | 92 | 64 | 25 | 133 | 91 | 155 |
| 0 | 90 | 67 | 91 | 155 | 1 | 68 |
| 0 | 1 | 155 | 91 | 67 | 90 | 155 |
| 0 | 91 | 133 | 25 | 64 | 92 | 67 |
| 0 | 91 | 65 | 92 | 66 | 90 | 156 |
| 107 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 41 | 131 | 69 | 76 | 29 | 88 |
| 0 | 155 | 44 | 18 | 68 | 117 | 88 |
| 0 | 150 | 65 | 69 | 7 | 91 | 91 |
| 0 | 45 | 132 | 109 | 109 | 95 | 88 |
| 0 | 34 | 43 | 69 | 28 | 120 | 139 |
| 0 | 107 | 135 | 69 | 116 | 26 | 88 |
G:=sub<GL(7,GF(157))| [1,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,144],[156,0,0,0,0,0,0,0,90,92,90,1,91,91,0,66,64,67,155,133,65,0,92,25,91,91,25,92,0,65,133,155,67,64,66,0,91,91,1,90,92,90,0,67,155,68,155,67,156],[107,0,0,0,0,0,0,0,41,155,150,45,34,107,0,131,44,65,132,43,135,0,69,18,69,109,69,69,0,76,68,7,109,28,116,0,29,117,91,95,120,26,0,88,88,91,88,139,88] >;
C3×C26.C6 in GAP, Magma, Sage, TeX
C_3\times C_{26}.C_6 % in TeX
G:=Group("C3xC26.C6"); // GroupNames label
G:=SmallGroup(468,19);
// by ID
G=gap.SmallGroup(468,19);
# by ID
G:=PCGroup([5,-2,-3,-3,-2,-13,90,10804,1359]);
// Polycyclic
G:=Group<a,b,c|a^3=b^26=1,c^6=b^13,a*b=b*a,a*c=c*a,c*b*c^-1=b^23>;
// generators/relations
Export