metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C39⋊7D4, D78⋊2C2, C2.5D78, C6.12D26, C26.12D6, C22⋊2D39, Dic39⋊1C2, C78.12C22, (C2×C26)⋊4S3, (C2×C78)⋊2C2, (C2×C6)⋊2D13, C13⋊3(C3⋊D4), C3⋊3(C13⋊D4), SmallGroup(312,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C39⋊7D4
G = < a,b,c | a39=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C39⋊7D4
►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)
(1 106 41 142)(2 105 42 141)(3 104 43 140)(4 103 44 139)(5 102 45 138)(6 101 46 137)(7 100 47 136)(8 99 48 135)(9 98 49 134)(10 97 50 133)(11 96 51 132)(12 95 52 131)(13 94 53 130)(14 93 54 129)(15 92 55 128)(16 91 56 127)(17 90 57 126)(18 89 58 125)(19 88 59 124)(20 87 60 123)(21 86 61 122)(22 85 62 121)(23 84 63 120)(24 83 64 119)(25 82 65 118)(26 81 66 156)(27 80 67 155)(28 79 68 154)(29 117 69 153)(30 116 70 152)(31 115 71 151)(32 114 72 150)(33 113 73 149)(34 112 74 148)(35 111 75 147)(36 110 76 146)(37 109 77 145)(38 108 78 144)(39 107 40 143)
(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(40 42)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(79 130)(80 129)(81 128)(82 127)(83 126)(84 125)(85 124)(86 123)(87 122)(88 121)(89 120)(90 119)(91 118)(92 156)(93 155)(94 154)(95 153)(96 152)(97 151)(98 150)(99 149)(100 148)(101 147)(102 146)(103 145)(104 144)(105 143)(106 142)(107 141)(108 140)(109 139)(110 138)(111 137)(112 136)(113 135)(114 134)(115 133)(116 132)(117 131)
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), (1,106,41,142)(2,105,42,141)(3,104,43,140)(4,103,44,139)(5,102,45,138)(6,101,46,137)(7,100,47,136)(8,99,48,135)(9,98,49,134)(10,97,50,133)(11,96,51,132)(12,95,52,131)(13,94,53,130)(14,93,54,129)(15,92,55,128)(16,91,56,127)(17,90,57,126)(18,89,58,125)(19,88,59,124)(20,87,60,123)(21,86,61,122)(22,85,62,121)(23,84,63,120)(24,83,64,119)(25,82,65,118)(26,81,66,156)(27,80,67,155)(28,79,68,154)(29,117,69,153)(30,116,70,152)(31,115,71,151)(32,114,72,150)(33,113,73,149)(34,112,74,148)(35,111,75,147)(36,110,76,146)(37,109,77,145)(38,108,78,144)(39,107,40,143), (2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(40,42)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(79,130)(80,129)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,120)(90,119)(91,118)(92,156)(93,155)(94,154)(95,153)(96,152)(97,151)(98,150)(99,149)(100,148)(101,147)(102,146)(103,145)(104,144)(105,143)(106,142)(107,141)(108,140)(109,139)(110,138)(111,137)(112,136)(113,135)(114,134)(115,133)(116,132)(117,131)>;
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), (1,106,41,142)(2,105,42,141)(3,104,43,140)(4,103,44,139)(5,102,45,138)(6,101,46,137)(7,100,47,136)(8,99,48,135)(9,98,49,134)(10,97,50,133)(11,96,51,132)(12,95,52,131)(13,94,53,130)(14,93,54,129)(15,92,55,128)(16,91,56,127)(17,90,57,126)(18,89,58,125)(19,88,59,124)(20,87,60,123)(21,86,61,122)(22,85,62,121)(23,84,63,120)(24,83,64,119)(25,82,65,118)(26,81,66,156)(27,80,67,155)(28,79,68,154)(29,117,69,153)(30,116,70,152)(31,115,71,151)(32,114,72,150)(33,113,73,149)(34,112,74,148)(35,111,75,147)(36,110,76,146)(37,109,77,145)(38,108,78,144)(39,107,40,143), (2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(40,42)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(79,130)(80,129)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,120)(90,119)(91,118)(92,156)(93,155)(94,154)(95,153)(96,152)(97,151)(98,150)(99,149)(100,148)(101,147)(102,146)(103,145)(104,144)(105,143)(106,142)(107,141)(108,140)(109,139)(110,138)(111,137)(112,136)(113,135)(114,134)(115,133)(116,132)(117,131) );
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)], [(1,106,41,142),(2,105,42,141),(3,104,43,140),(4,103,44,139),(5,102,45,138),(6,101,46,137),(7,100,47,136),(8,99,48,135),(9,98,49,134),(10,97,50,133),(11,96,51,132),(12,95,52,131),(13,94,53,130),(14,93,54,129),(15,92,55,128),(16,91,56,127),(17,90,57,126),(18,89,58,125),(19,88,59,124),(20,87,60,123),(21,86,61,122),(22,85,62,121),(23,84,63,120),(24,83,64,119),(25,82,65,118),(26,81,66,156),(27,80,67,155),(28,79,68,154),(29,117,69,153),(30,116,70,152),(31,115,71,151),(32,114,72,150),(33,113,73,149),(34,112,74,148),(35,111,75,147),(36,110,76,146),(37,109,77,145),(38,108,78,144),(39,107,40,143)], [(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(40,42),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61),(79,130),(80,129),(81,128),(82,127),(83,126),(84,125),(85,124),(86,123),(87,122),(88,121),(89,120),(90,119),(91,118),(92,156),(93,155),(94,154),(95,153),(96,152),(97,151),(98,150),(99,149),(100,148),(101,147),(102,146),(103,145),(104,144),(105,143),(106,142),(107,141),(108,140),(109,139),(110,138),(111,137),(112,136),(113,135),(114,134),(115,133),(116,132),(117,131)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 13A | ··· | 13F | 26A | ··· | 26R | 39A | ··· | 39L | 78A | ··· | 78AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 2 | 78 | 2 | 78 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | C3⋊D4 | D13 | D26 | D39 | C13⋊D4 | D78 | C39⋊7D4 |
| kernel | C39⋊7D4 | Dic39 | D78 | C2×C78 | C2×C26 | C39 | C26 | C13 | C2×C6 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 12 | 12 | 12 | 24 |
Matrix representation of C39⋊7D4 ►in GL4(𝔽157) generated by
| 11 | 75 | 0 | 0 |
| 82 | 131 | 0 | 0 |
| 0 | 0 | 109 | 20 |
| 0 | 0 | 137 | 77 |
| 15 | 34 | 0 | 0 |
| 58 | 142 | 0 | 0 |
| 0 | 0 | 31 | 97 |
| 0 | 0 | 16 | 126 |
| 1 | 124 | 0 | 0 |
| 0 | 156 | 0 | 0 |
| 0 | 0 | 31 | 97 |
| 0 | 0 | 16 | 126 |
G:=sub<GL(4,GF(157))| [11,82,0,0,75,131,0,0,0,0,109,137,0,0,20,77],[15,58,0,0,34,142,0,0,0,0,31,16,0,0,97,126],[1,0,0,0,124,156,0,0,0,0,31,16,0,0,97,126] >;
C39⋊7D4 in GAP, Magma, Sage, TeX
C_{39}\rtimes_7D_4 % in TeX
G:=Group("C39:7D4"); // GroupNames label
G:=SmallGroup(312,41);
// by ID
G=gap.SmallGroup(312,41);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,61,323,7204]);
// Polycyclic
G:=Group<a,b,c|a^39=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export