metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C39⋊1D4, D6⋊1D13, D26⋊1S3, C6.4D26, C26.4D6, Dic39⋊4C2, C78.4C22, (S3×C26)⋊1C2, (C6×D13)⋊1C2, C13⋊2(C3⋊D4), C3⋊2(C13⋊D4), C2.4(S3×D13), SmallGroup(312,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C39⋊D4
G = < a,b,c | a39=b4=c2=1, bab-1=a-1, cac=a25, cbc=b-1 >
Smallest permutation representation of C39⋊D4
►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 83 51 138)(2 82 52 137)(3 81 53 136)(4 80 54 135)(5 79 55 134)(6 117 56 133)(7 116 57 132)(8 115 58 131)(9 114 59 130)(10 113 60 129)(11 112 61 128)(12 111 62 127)(13 110 63 126)(14 109 64 125)(15 108 65 124)(16 107 66 123)(17 106 67 122)(18 105 68 121)(19 104 69 120)(20 103 70 119)(21 102 71 118)(22 101 72 156)(23 100 73 155)(24 99 74 154)(25 98 75 153)(26 97 76 152)(27 96 77 151)(28 95 78 150)(29 94 40 149)(30 93 41 148)(31 92 42 147)(32 91 43 146)(33 90 44 145)(34 89 45 144)(35 88 46 143)(36 87 47 142)(37 86 48 141)(38 85 49 140)(39 84 50 139)
(2 26)(3 12)(4 37)(5 23)(6 9)(7 34)(8 20)(10 31)(11 17)(13 28)(15 39)(16 25)(18 36)(19 22)(21 33)(24 30)(29 38)(32 35)(40 49)(41 74)(42 60)(43 46)(44 71)(45 57)(47 68)(48 54)(50 65)(52 76)(53 62)(55 73)(56 59)(58 70)(61 67)(63 78)(66 75)(69 72)(79 155)(80 141)(81 127)(82 152)(83 138)(84 124)(85 149)(86 135)(87 121)(88 146)(89 132)(90 118)(91 143)(92 129)(93 154)(94 140)(95 126)(96 151)(97 137)(98 123)(99 148)(100 134)(101 120)(102 145)(103 131)(104 156)(105 142)(106 128)(107 153)(108 139)(109 125)(110 150)(111 136)(112 122)(113 147)(114 133)(115 119)(116 144)(117 130)
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), (1,83,51,138)(2,82,52,137)(3,81,53,136)(4,80,54,135)(5,79,55,134)(6,117,56,133)(7,116,57,132)(8,115,58,131)(9,114,59,130)(10,113,60,129)(11,112,61,128)(12,111,62,127)(13,110,63,126)(14,109,64,125)(15,108,65,124)(16,107,66,123)(17,106,67,122)(18,105,68,121)(19,104,69,120)(20,103,70,119)(21,102,71,118)(22,101,72,156)(23,100,73,155)(24,99,74,154)(25,98,75,153)(26,97,76,152)(27,96,77,151)(28,95,78,150)(29,94,40,149)(30,93,41,148)(31,92,42,147)(32,91,43,146)(33,90,44,145)(34,89,45,144)(35,88,46,143)(36,87,47,142)(37,86,48,141)(38,85,49,140)(39,84,50,139), (2,26)(3,12)(4,37)(5,23)(6,9)(7,34)(8,20)(10,31)(11,17)(13,28)(15,39)(16,25)(18,36)(19,22)(21,33)(24,30)(29,38)(32,35)(40,49)(41,74)(42,60)(43,46)(44,71)(45,57)(47,68)(48,54)(50,65)(52,76)(53,62)(55,73)(56,59)(58,70)(61,67)(63,78)(66,75)(69,72)(79,155)(80,141)(81,127)(82,152)(83,138)(84,124)(85,149)(86,135)(87,121)(88,146)(89,132)(90,118)(91,143)(92,129)(93,154)(94,140)(95,126)(96,151)(97,137)(98,123)(99,148)(100,134)(101,120)(102,145)(103,131)(104,156)(105,142)(106,128)(107,153)(108,139)(109,125)(110,150)(111,136)(112,122)(113,147)(114,133)(115,119)(116,144)(117,130)>;
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), (1,83,51,138)(2,82,52,137)(3,81,53,136)(4,80,54,135)(5,79,55,134)(6,117,56,133)(7,116,57,132)(8,115,58,131)(9,114,59,130)(10,113,60,129)(11,112,61,128)(12,111,62,127)(13,110,63,126)(14,109,64,125)(15,108,65,124)(16,107,66,123)(17,106,67,122)(18,105,68,121)(19,104,69,120)(20,103,70,119)(21,102,71,118)(22,101,72,156)(23,100,73,155)(24,99,74,154)(25,98,75,153)(26,97,76,152)(27,96,77,151)(28,95,78,150)(29,94,40,149)(30,93,41,148)(31,92,42,147)(32,91,43,146)(33,90,44,145)(34,89,45,144)(35,88,46,143)(36,87,47,142)(37,86,48,141)(38,85,49,140)(39,84,50,139), (2,26)(3,12)(4,37)(5,23)(6,9)(7,34)(8,20)(10,31)(11,17)(13,28)(15,39)(16,25)(18,36)(19,22)(21,33)(24,30)(29,38)(32,35)(40,49)(41,74)(42,60)(43,46)(44,71)(45,57)(47,68)(48,54)(50,65)(52,76)(53,62)(55,73)(56,59)(58,70)(61,67)(63,78)(66,75)(69,72)(79,155)(80,141)(81,127)(82,152)(83,138)(84,124)(85,149)(86,135)(87,121)(88,146)(89,132)(90,118)(91,143)(92,129)(93,154)(94,140)(95,126)(96,151)(97,137)(98,123)(99,148)(100,134)(101,120)(102,145)(103,131)(104,156)(105,142)(106,128)(107,153)(108,139)(109,125)(110,150)(111,136)(112,122)(113,147)(114,133)(115,119)(116,144)(117,130) );
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)], [(1,83,51,138),(2,82,52,137),(3,81,53,136),(4,80,54,135),(5,79,55,134),(6,117,56,133),(7,116,57,132),(8,115,58,131),(9,114,59,130),(10,113,60,129),(11,112,61,128),(12,111,62,127),(13,110,63,126),(14,109,64,125),(15,108,65,124),(16,107,66,123),(17,106,67,122),(18,105,68,121),(19,104,69,120),(20,103,70,119),(21,102,71,118),(22,101,72,156),(23,100,73,155),(24,99,74,154),(25,98,75,153),(26,97,76,152),(27,96,77,151),(28,95,78,150),(29,94,40,149),(30,93,41,148),(31,92,42,147),(32,91,43,146),(33,90,44,145),(34,89,45,144),(35,88,46,143),(36,87,47,142),(37,86,48,141),(38,85,49,140),(39,84,50,139)], [(2,26),(3,12),(4,37),(5,23),(6,9),(7,34),(8,20),(10,31),(11,17),(13,28),(15,39),(16,25),(18,36),(19,22),(21,33),(24,30),(29,38),(32,35),(40,49),(41,74),(42,60),(43,46),(44,71),(45,57),(47,68),(48,54),(50,65),(52,76),(53,62),(55,73),(56,59),(58,70),(61,67),(63,78),(66,75),(69,72),(79,155),(80,141),(81,127),(82,152),(83,138),(84,124),(85,149),(86,135),(87,121),(88,146),(89,132),(90,118),(91,143),(92,129),(93,154),(94,140),(95,126),(96,151),(97,137),(98,123),(99,148),(100,134),(101,120),(102,145),(103,131),(104,156),(105,142),(106,128),(107,153),(108,139),(109,125),(110,150),(111,136),(112,122),(113,147),(114,133),(115,119),(116,144),(117,130)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26R | 39A | ··· | 39F | 78A | ··· | 78F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 6 | 26 | 2 | 78 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | C3⋊D4 | D13 | D26 | C13⋊D4 | S3×D13 | C39⋊D4 |
| kernel | C39⋊D4 | Dic39 | C6×D13 | S3×C26 | D26 | C39 | C26 | C13 | D6 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 12 | 6 | 6 |
Matrix representation of C39⋊D4 ►in GL4(𝔽157) generated by
| 52 | 134 | 0 | 0 |
| 23 | 14 | 0 | 0 |
| 0 | 0 | 155 | 93 |
| 0 | 0 | 81 | 1 |
| 139 | 154 | 0 | 0 |
| 56 | 18 | 0 | 0 |
| 0 | 0 | 66 | 43 |
| 0 | 0 | 147 | 91 |
| 1 | 0 | 0 | 0 |
| 145 | 156 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(157))| [52,23,0,0,134,14,0,0,0,0,155,81,0,0,93,1],[139,56,0,0,154,18,0,0,0,0,66,147,0,0,43,91],[1,145,0,0,0,156,0,0,0,0,1,0,0,0,0,1] >;
C39⋊D4 in GAP, Magma, Sage, TeX
C_{39}\rtimes D_4 % in TeX
G:=Group("C39:D4"); // GroupNames label
G:=SmallGroup(312,18);
// by ID
G=gap.SmallGroup(312,18);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,61,168,7204]);
// Polycyclic
G:=Group<a,b,c|a^39=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^25,c*b*c=b^-1>;
// generators/relations
Export