direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C2×C26, C39⋊4C23, C78⋊4C22, C6⋊(C2×C26), C3⋊(C22×C26), (C2×C6)⋊3C26, (C2×C78)⋊7C2, SmallGroup(312,59)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C26 |
Generators and relations for S3×C2×C26
G = < a,b,c,d | a2=b26=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 108 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C23, D6, C2×C6, C13, C22×S3, C26, C26, C39, C2×C26, C2×C26, S3×C13, C78, C22×C26, S3×C26, C2×C78, S3×C2×C26
Quotients: C1, C2, C22, S3, C23, D6, C13, C22×S3, C26, C2×C26, S3×C13, C22×C26, S3×C26, S3×C2×C26
►On 156 points
Generators in S156
(1 133)(2 134)(3 135)(4 136)(5 137)(6 138)(7 139)(8 140)(9 141)(10 142)(11 143)(12 144)(13 145)(14 146)(15 147)(16 148)(17 149)(18 150)(19 151)(20 152)(21 153)(22 154)(23 155)(24 156)(25 131)(26 132)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 53)(45 54)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)(101 105)(102 106)(103 107)(104 108)
(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 91 36)(2 92 37)(3 93 38)(4 94 39)(5 95 40)(6 96 41)(7 97 42)(8 98 43)(9 99 44)(10 100 45)(11 101 46)(12 102 47)(13 103 48)(14 104 49)(15 79 50)(16 80 51)(17 81 52)(18 82 27)(19 83 28)(20 84 29)(21 85 30)(22 86 31)(23 87 32)(24 88 33)(25 89 34)(26 90 35)(53 141 129)(54 142 130)(55 143 105)(56 144 106)(57 145 107)(58 146 108)(59 147 109)(60 148 110)(61 149 111)(62 150 112)(63 151 113)(64 152 114)(65 153 115)(66 154 116)(67 155 117)(68 156 118)(69 131 119)(70 132 120)(71 133 121)(72 134 122)(73 135 123)(74 136 124)(75 137 125)(76 138 126)(77 139 127)(78 140 128)
(1 133)(2 134)(3 135)(4 136)(5 137)(6 138)(7 139)(8 140)(9 141)(10 142)(11 143)(12 144)(13 145)(14 146)(15 147)(16 148)(17 149)(18 150)(19 151)(20 152)(21 153)(22 154)(23 155)(24 156)(25 131)(26 132)(27 112)(28 113)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 120)(36 121)(37 122)(38 123)(39 124)(40 125)(41 126)(42 127)(43 128)(44 129)(45 130)(46 105)(47 106)(48 107)(49 108)(50 109)(51 110)(52 111)(53 99)(54 100)(55 101)(56 102)(57 103)(58 104)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)
G:=sub<Sym(156)| (1,133)(2,134)(3,135)(4,136)(5,137)(6,138)(7,139)(8,140)(9,141)(10,142)(11,143)(12,144)(13,145)(14,146)(15,147)(16,148)(17,149)(18,150)(19,151)(20,152)(21,153)(22,154)(23,155)(24,156)(25,131)(26,132)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,53)(45,54)(46,55)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(98,128)(99,129)(100,130)(101,105)(102,106)(103,107)(104,108), (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), (1,91,36)(2,92,37)(3,93,38)(4,94,39)(5,95,40)(6,96,41)(7,97,42)(8,98,43)(9,99,44)(10,100,45)(11,101,46)(12,102,47)(13,103,48)(14,104,49)(15,79,50)(16,80,51)(17,81,52)(18,82,27)(19,83,28)(20,84,29)(21,85,30)(22,86,31)(23,87,32)(24,88,33)(25,89,34)(26,90,35)(53,141,129)(54,142,130)(55,143,105)(56,144,106)(57,145,107)(58,146,108)(59,147,109)(60,148,110)(61,149,111)(62,150,112)(63,151,113)(64,152,114)(65,153,115)(66,154,116)(67,155,117)(68,156,118)(69,131,119)(70,132,120)(71,133,121)(72,134,122)(73,135,123)(74,136,124)(75,137,125)(76,138,126)(77,139,127)(78,140,128), (1,133)(2,134)(3,135)(4,136)(5,137)(6,138)(7,139)(8,140)(9,141)(10,142)(11,143)(12,144)(13,145)(14,146)(15,147)(16,148)(17,149)(18,150)(19,151)(20,152)(21,153)(22,154)(23,155)(24,156)(25,131)(26,132)(27,112)(28,113)(29,114)(30,115)(31,116)(32,117)(33,118)(34,119)(35,120)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,129)(45,130)(46,105)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,99)(54,100)(55,101)(56,102)(57,103)(58,104)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)>;
G:=Group( (1,133)(2,134)(3,135)(4,136)(5,137)(6,138)(7,139)(8,140)(9,141)(10,142)(11,143)(12,144)(13,145)(14,146)(15,147)(16,148)(17,149)(18,150)(19,151)(20,152)(21,153)(22,154)(23,155)(24,156)(25,131)(26,132)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,53)(45,54)(46,55)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(98,128)(99,129)(100,130)(101,105)(102,106)(103,107)(104,108), (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), (1,91,36)(2,92,37)(3,93,38)(4,94,39)(5,95,40)(6,96,41)(7,97,42)(8,98,43)(9,99,44)(10,100,45)(11,101,46)(12,102,47)(13,103,48)(14,104,49)(15,79,50)(16,80,51)(17,81,52)(18,82,27)(19,83,28)(20,84,29)(21,85,30)(22,86,31)(23,87,32)(24,88,33)(25,89,34)(26,90,35)(53,141,129)(54,142,130)(55,143,105)(56,144,106)(57,145,107)(58,146,108)(59,147,109)(60,148,110)(61,149,111)(62,150,112)(63,151,113)(64,152,114)(65,153,115)(66,154,116)(67,155,117)(68,156,118)(69,131,119)(70,132,120)(71,133,121)(72,134,122)(73,135,123)(74,136,124)(75,137,125)(76,138,126)(77,139,127)(78,140,128), (1,133)(2,134)(3,135)(4,136)(5,137)(6,138)(7,139)(8,140)(9,141)(10,142)(11,143)(12,144)(13,145)(14,146)(15,147)(16,148)(17,149)(18,150)(19,151)(20,152)(21,153)(22,154)(23,155)(24,156)(25,131)(26,132)(27,112)(28,113)(29,114)(30,115)(31,116)(32,117)(33,118)(34,119)(35,120)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,129)(45,130)(46,105)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,99)(54,100)(55,101)(56,102)(57,103)(58,104)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98) );
G=PermutationGroup([[(1,133),(2,134),(3,135),(4,136),(5,137),(6,138),(7,139),(8,140),(9,141),(10,142),(11,143),(12,144),(13,145),(14,146),(15,147),(16,148),(17,149),(18,150),(19,151),(20,152),(21,153),(22,154),(23,155),(24,156),(25,131),(26,132),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71),(37,72),(38,73),(39,74),(40,75),(41,76),(42,77),(43,78),(44,53),(45,54),(46,55),(47,56),(48,57),(49,58),(50,59),(51,60),(52,61),(79,109),(80,110),(81,111),(82,112),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120),(91,121),(92,122),(93,123),(94,124),(95,125),(96,126),(97,127),(98,128),(99,129),(100,130),(101,105),(102,106),(103,107),(104,108)], [(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)], [(1,91,36),(2,92,37),(3,93,38),(4,94,39),(5,95,40),(6,96,41),(7,97,42),(8,98,43),(9,99,44),(10,100,45),(11,101,46),(12,102,47),(13,103,48),(14,104,49),(15,79,50),(16,80,51),(17,81,52),(18,82,27),(19,83,28),(20,84,29),(21,85,30),(22,86,31),(23,87,32),(24,88,33),(25,89,34),(26,90,35),(53,141,129),(54,142,130),(55,143,105),(56,144,106),(57,145,107),(58,146,108),(59,147,109),(60,148,110),(61,149,111),(62,150,112),(63,151,113),(64,152,114),(65,153,115),(66,154,116),(67,155,117),(68,156,118),(69,131,119),(70,132,120),(71,133,121),(72,134,122),(73,135,123),(74,136,124),(75,137,125),(76,138,126),(77,139,127),(78,140,128)], [(1,133),(2,134),(3,135),(4,136),(5,137),(6,138),(7,139),(8,140),(9,141),(10,142),(11,143),(12,144),(13,145),(14,146),(15,147),(16,148),(17,149),(18,150),(19,151),(20,152),(21,153),(22,154),(23,155),(24,156),(25,131),(26,132),(27,112),(28,113),(29,114),(30,115),(31,116),(32,117),(33,118),(34,119),(35,120),(36,121),(37,122),(38,123),(39,124),(40,125),(41,126),(42,127),(43,128),(44,129),(45,130),(46,105),(47,106),(48,107),(49,108),(50,109),(51,110),(52,111),(53,99),(54,100),(55,101),(56,102),(57,103),(58,104),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,91),(72,92),(73,93),(74,94),(75,95),(76,96),(77,97),(78,98)]])
156 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 6A | 6B | 6C | 13A | ··· | 13L | 26A | ··· | 26AJ | 26AK | ··· | 26CF | 39A | ··· | 39L | 78A | ··· | 78AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | S3 | D6 | S3×C13 | S3×C26 |
| kernel | S3×C2×C26 | S3×C26 | C2×C78 | C22×S3 | D6 | C2×C6 | C2×C26 | C26 | C22 | C2 |
| # reps | 1 | 6 | 1 | 12 | 72 | 12 | 1 | 3 | 12 | 36 |
Matrix representation of S3×C2×C26 ►in GL4(𝔽79) generated by
| 78 | 0 | 0 | 0 |
| 0 | 78 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 78 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 38 | 0 |
| 0 | 0 | 0 | 38 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 78 | 78 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 78 | 78 |
G:=sub<GL(4,GF(79))| [78,0,0,0,0,78,0,0,0,0,1,0,0,0,0,1],[78,0,0,0,0,1,0,0,0,0,38,0,0,0,0,38],[1,0,0,0,0,1,0,0,0,0,78,1,0,0,78,0],[1,0,0,0,0,1,0,0,0,0,1,78,0,0,0,78] >;
S3×C2×C26 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_{26} % in TeX
G:=Group("S3xC2xC26"); // GroupNames label
G:=SmallGroup(312,59);
// by ID
G=gap.SmallGroup(312,59);
# by ID
G:=PCGroup([5,-2,-2,-2,-13,-3,5204]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^26=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations