metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D78.C2, D39⋊2C4, C6.3D26, C26.3D6, Dic13⋊2S3, Dic3⋊2D13, C78.3C22, C13⋊2(C4×S3), C39⋊6(C2×C4), C3⋊1(C4×D13), C2.3(S3×D13), (C3×Dic13)⋊2C2, (Dic3×C13)⋊2C2, SmallGroup(312,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — D78.C2 |
Generators and relations for D78.C2
G = < a,b,c | a78=b2=1, c2=a39, bab=a-1, cac-1=a25, cbc-1=a24b >
Smallest permutation representation of D78.C2
►On 156 points
Generators in S156
(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 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(79 124)(80 123)(81 122)(82 121)(83 120)(84 119)(85 118)(86 117)(87 116)(88 115)(89 114)(90 113)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(101 102)(125 156)(126 155)(127 154)(128 153)(129 152)(130 151)(131 150)(132 149)(133 148)(134 147)(135 146)(136 145)(137 144)(138 143)(139 142)(140 141)
(1 102 40 141)(2 127 41 88)(3 152 42 113)(4 99 43 138)(5 124 44 85)(6 149 45 110)(7 96 46 135)(8 121 47 82)(9 146 48 107)(10 93 49 132)(11 118 50 79)(12 143 51 104)(13 90 52 129)(14 115 53 154)(15 140 54 101)(16 87 55 126)(17 112 56 151)(18 137 57 98)(19 84 58 123)(20 109 59 148)(21 134 60 95)(22 81 61 120)(23 106 62 145)(24 131 63 92)(25 156 64 117)(26 103 65 142)(27 128 66 89)(28 153 67 114)(29 100 68 139)(30 125 69 86)(31 150 70 111)(32 97 71 136)(33 122 72 83)(34 147 73 108)(35 94 74 133)(36 119 75 80)(37 144 76 105)(38 91 77 130)(39 116 78 155)
G:=sub<Sym(156)| (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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(79,124)(80,123)(81,122)(82,121)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)(125,156)(126,155)(127,154)(128,153)(129,152)(130,151)(131,150)(132,149)(133,148)(134,147)(135,146)(136,145)(137,144)(138,143)(139,142)(140,141), (1,102,40,141)(2,127,41,88)(3,152,42,113)(4,99,43,138)(5,124,44,85)(6,149,45,110)(7,96,46,135)(8,121,47,82)(9,146,48,107)(10,93,49,132)(11,118,50,79)(12,143,51,104)(13,90,52,129)(14,115,53,154)(15,140,54,101)(16,87,55,126)(17,112,56,151)(18,137,57,98)(19,84,58,123)(20,109,59,148)(21,134,60,95)(22,81,61,120)(23,106,62,145)(24,131,63,92)(25,156,64,117)(26,103,65,142)(27,128,66,89)(28,153,67,114)(29,100,68,139)(30,125,69,86)(31,150,70,111)(32,97,71,136)(33,122,72,83)(34,147,73,108)(35,94,74,133)(36,119,75,80)(37,144,76,105)(38,91,77,130)(39,116,78,155)>;
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)(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(79,124)(80,123)(81,122)(82,121)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)(125,156)(126,155)(127,154)(128,153)(129,152)(130,151)(131,150)(132,149)(133,148)(134,147)(135,146)(136,145)(137,144)(138,143)(139,142)(140,141), (1,102,40,141)(2,127,41,88)(3,152,42,113)(4,99,43,138)(5,124,44,85)(6,149,45,110)(7,96,46,135)(8,121,47,82)(9,146,48,107)(10,93,49,132)(11,118,50,79)(12,143,51,104)(13,90,52,129)(14,115,53,154)(15,140,54,101)(16,87,55,126)(17,112,56,151)(18,137,57,98)(19,84,58,123)(20,109,59,148)(21,134,60,95)(22,81,61,120)(23,106,62,145)(24,131,63,92)(25,156,64,117)(26,103,65,142)(27,128,66,89)(28,153,67,114)(29,100,68,139)(30,125,69,86)(31,150,70,111)(32,97,71,136)(33,122,72,83)(34,147,73,108)(35,94,74,133)(36,119,75,80)(37,144,76,105)(38,91,77,130)(39,116,78,155) );
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),(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)], [(1,78),(2,77),(3,76),(4,75),(5,74),(6,73),(7,72),(8,71),(9,70),(10,69),(11,68),(12,67),(13,66),(14,65),(15,64),(16,63),(17,62),(18,61),(19,60),(20,59),(21,58),(22,57),(23,56),(24,55),(25,54),(26,53),(27,52),(28,51),(29,50),(30,49),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(79,124),(80,123),(81,122),(82,121),(83,120),(84,119),(85,118),(86,117),(87,116),(88,115),(89,114),(90,113),(91,112),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103),(101,102),(125,156),(126,155),(127,154),(128,153),(129,152),(130,151),(131,150),(132,149),(133,148),(134,147),(135,146),(136,145),(137,144),(138,143),(139,142),(140,141)], [(1,102,40,141),(2,127,41,88),(3,152,42,113),(4,99,43,138),(5,124,44,85),(6,149,45,110),(7,96,46,135),(8,121,47,82),(9,146,48,107),(10,93,49,132),(11,118,50,79),(12,143,51,104),(13,90,52,129),(14,115,53,154),(15,140,54,101),(16,87,55,126),(17,112,56,151),(18,137,57,98),(19,84,58,123),(20,109,59,148),(21,134,60,95),(22,81,61,120),(23,106,62,145),(24,131,63,92),(25,156,64,117),(26,103,65,142),(27,128,66,89),(28,153,67,114),(29,100,68,139),(30,125,69,86),(31,150,70,111),(32,97,71,136),(33,122,72,83),(34,147,73,108),(35,94,74,133),(36,119,75,80),(37,144,76,105),(38,91,77,130),(39,116,78,155)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 12A | 12B | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39F | 52A | ··· | 52L | 78A | ··· | 78F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 |
| size | 1 | 1 | 39 | 39 | 2 | 3 | 3 | 13 | 13 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | C4×S3 | D13 | D26 | C4×D13 | S3×D13 | D78.C2 |
| kernel | D78.C2 | Dic3×C13 | C3×Dic13 | D78 | D39 | Dic13 | C26 | C13 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 6 | 6 | 12 | 6 | 6 |
Matrix representation of D78.C2 ►in GL4(𝔽157) generated by
| 66 | 91 | 0 | 0 |
| 93 | 133 | 0 | 0 |
| 0 | 0 | 1 | 17 |
| 0 | 0 | 46 | 155 |
| 115 | 6 | 0 | 0 |
| 151 | 42 | 0 | 0 |
| 0 | 0 | 1 | 17 |
| 0 | 0 | 0 | 156 |
| 117 | 117 | 0 | 0 |
| 146 | 40 | 0 | 0 |
| 0 | 0 | 156 | 0 |
| 0 | 0 | 0 | 156 |
G:=sub<GL(4,GF(157))| [66,93,0,0,91,133,0,0,0,0,1,46,0,0,17,155],[115,151,0,0,6,42,0,0,0,0,1,0,0,0,17,156],[117,146,0,0,117,40,0,0,0,0,156,0,0,0,0,156] >;
D78.C2 in GAP, Magma, Sage, TeX
D_{78}.C_2 % in TeX
G:=Group("D78.C2"); // GroupNames label
G:=SmallGroup(312,17);
// by ID
G=gap.SmallGroup(312,17);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,20,26,168,7204]);
// Polycyclic
G:=Group<a,b,c|a^78=b^2=1,c^2=a^39,b*a*b=a^-1,c*a*c^-1=a^25,c*b*c^-1=a^24*b>;
// generators/relations
Export