direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C2×C79⋊C3, C158⋊C3, C79⋊2C6, SmallGroup(474,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C79 — C2×C79⋊C3 |
Generators and relations for C2×C79⋊C3
G = < a,b,c | a2=b79=c3=1, ab=ba, ac=ca, cbc-1=b55 >
Smallest permutation representation of C2×C79⋊C3
►On 158 points
Generators in S158
(1 80)(2 81)(3 82)(4 83)(5 84)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 91)(13 92)(14 93)(15 94)(16 95)(17 96)(18 97)(19 98)(20 99)(21 100)(22 101)(23 102)(24 103)(25 104)(26 105)(27 106)(28 107)(29 108)(30 109)(31 110)(32 111)(33 112)(34 113)(35 114)(36 115)(37 116)(38 117)(39 118)(40 119)(41 120)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(49 128)(50 129)(51 130)(52 131)(53 132)(54 133)(55 134)(56 135)(57 136)(58 137)(59 138)(60 139)(61 140)(62 141)(63 142)(64 143)(65 144)(66 145)(67 146)(68 147)(69 148)(70 149)(71 150)(72 151)(73 152)(74 153)(75 154)(76 155)(77 156)(78 157)(79 158)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158)
(2 24 56)(3 47 32)(4 70 8)(5 14 63)(6 37 39)(7 60 15)(9 27 46)(10 50 22)(11 73 77)(12 17 53)(13 40 29)(16 30 36)(18 76 67)(19 20 43)(21 66 74)(23 33 26)(25 79 57)(28 69 64)(31 59 71)(34 49 78)(35 72 54)(38 62 61)(41 52 68)(42 75 44)(45 65 51)(48 55 58)(81 103 135)(82 126 111)(83 149 87)(84 93 142)(85 116 118)(86 139 94)(88 106 125)(89 129 101)(90 152 156)(91 96 132)(92 119 108)(95 109 115)(97 155 146)(98 99 122)(100 145 153)(102 112 105)(104 158 136)(107 148 143)(110 138 150)(113 128 157)(114 151 133)(117 141 140)(120 131 147)(121 154 123)(124 144 130)(127 134 137)
G:=sub<Sym(158)| (1,80)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,96)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(41,120)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(49,128)(50,129)(51,130)(52,131)(53,132)(54,133)(55,134)(56,135)(57,136)(58,137)(59,138)(60,139)(61,140)(62,141)(63,142)(64,143)(65,144)(66,145)(67,146)(68,147)(69,148)(70,149)(71,150)(72,151)(73,152)(74,153)(75,154)(76,155)(77,156)(78,157)(79,158), (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)(80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158), (2,24,56)(3,47,32)(4,70,8)(5,14,63)(6,37,39)(7,60,15)(9,27,46)(10,50,22)(11,73,77)(12,17,53)(13,40,29)(16,30,36)(18,76,67)(19,20,43)(21,66,74)(23,33,26)(25,79,57)(28,69,64)(31,59,71)(34,49,78)(35,72,54)(38,62,61)(41,52,68)(42,75,44)(45,65,51)(48,55,58)(81,103,135)(82,126,111)(83,149,87)(84,93,142)(85,116,118)(86,139,94)(88,106,125)(89,129,101)(90,152,156)(91,96,132)(92,119,108)(95,109,115)(97,155,146)(98,99,122)(100,145,153)(102,112,105)(104,158,136)(107,148,143)(110,138,150)(113,128,157)(114,151,133)(117,141,140)(120,131,147)(121,154,123)(124,144,130)(127,134,137)>;
G:=Group( (1,80)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,96)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(41,120)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(49,128)(50,129)(51,130)(52,131)(53,132)(54,133)(55,134)(56,135)(57,136)(58,137)(59,138)(60,139)(61,140)(62,141)(63,142)(64,143)(65,144)(66,145)(67,146)(68,147)(69,148)(70,149)(71,150)(72,151)(73,152)(74,153)(75,154)(76,155)(77,156)(78,157)(79,158), (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)(80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158), (2,24,56)(3,47,32)(4,70,8)(5,14,63)(6,37,39)(7,60,15)(9,27,46)(10,50,22)(11,73,77)(12,17,53)(13,40,29)(16,30,36)(18,76,67)(19,20,43)(21,66,74)(23,33,26)(25,79,57)(28,69,64)(31,59,71)(34,49,78)(35,72,54)(38,62,61)(41,52,68)(42,75,44)(45,65,51)(48,55,58)(81,103,135)(82,126,111)(83,149,87)(84,93,142)(85,116,118)(86,139,94)(88,106,125)(89,129,101)(90,152,156)(91,96,132)(92,119,108)(95,109,115)(97,155,146)(98,99,122)(100,145,153)(102,112,105)(104,158,136)(107,148,143)(110,138,150)(113,128,157)(114,151,133)(117,141,140)(120,131,147)(121,154,123)(124,144,130)(127,134,137) );
G=PermutationGroup([[(1,80),(2,81),(3,82),(4,83),(5,84),(6,85),(7,86),(8,87),(9,88),(10,89),(11,90),(12,91),(13,92),(14,93),(15,94),(16,95),(17,96),(18,97),(19,98),(20,99),(21,100),(22,101),(23,102),(24,103),(25,104),(26,105),(27,106),(28,107),(29,108),(30,109),(31,110),(32,111),(33,112),(34,113),(35,114),(36,115),(37,116),(38,117),(39,118),(40,119),(41,120),(42,121),(43,122),(44,123),(45,124),(46,125),(47,126),(48,127),(49,128),(50,129),(51,130),(52,131),(53,132),(54,133),(55,134),(56,135),(57,136),(58,137),(59,138),(60,139),(61,140),(62,141),(63,142),(64,143),(65,144),(66,145),(67,146),(68,147),(69,148),(70,149),(71,150),(72,151),(73,152),(74,153),(75,154),(76,155),(77,156),(78,157),(79,158)], [(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),(80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158)], [(2,24,56),(3,47,32),(4,70,8),(5,14,63),(6,37,39),(7,60,15),(9,27,46),(10,50,22),(11,73,77),(12,17,53),(13,40,29),(16,30,36),(18,76,67),(19,20,43),(21,66,74),(23,33,26),(25,79,57),(28,69,64),(31,59,71),(34,49,78),(35,72,54),(38,62,61),(41,52,68),(42,75,44),(45,65,51),(48,55,58),(81,103,135),(82,126,111),(83,149,87),(84,93,142),(85,116,118),(86,139,94),(88,106,125),(89,129,101),(90,152,156),(91,96,132),(92,119,108),(95,109,115),(97,155,146),(98,99,122),(100,145,153),(102,112,105),(104,158,136),(107,148,143),(110,138,150),(113,128,157),(114,151,133),(117,141,140),(120,131,147),(121,154,123),(124,144,130),(127,134,137)]])
58 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 79A | ··· | 79Z | 158A | ··· | 158Z |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 79 | ··· | 79 | 158 | ··· | 158 |
| size | 1 | 1 | 79 | 79 | 79 | 79 | 3 | ··· | 3 | 3 | ··· | 3 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C2 | C3 | C6 | C79⋊C3 | C2×C79⋊C3 |
| kernel | C2×C79⋊C3 | C79⋊C3 | C158 | C79 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 26 | 26 |
Matrix representation of C2×C79⋊C3 ►in GL3(𝔽1423) generated by
| 1422 | 0 | 0 |
| 0 | 1422 | 0 |
| 0 | 0 | 1422 |
| 541 | 251 | 1 |
| 51 | 544 | 1068 |
| 302 | 270 | 1271 |
| 1194 | 374 | 1175 |
| 356 | 962 | 725 |
| 1202 | 56 | 690 |
G:=sub<GL(3,GF(1423))| [1422,0,0,0,1422,0,0,0,1422],[541,51,302,251,544,270,1,1068,1271],[1194,356,1202,374,962,56,1175,725,690] >;
C2×C79⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_{79}\rtimes C_3 % in TeX
G:=Group("C2xC79:C3"); // GroupNames label
G:=SmallGroup(474,2);
// by ID
G=gap.SmallGroup(474,2);
# by ID
G:=PCGroup([3,-2,-3,-79,626]);
// Polycyclic
G:=Group<a,b,c|a^2=b^79=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^55>;
// generators/relations
Export