direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C2×C73⋊C3, C146⋊C3, C73⋊2C6, SmallGroup(438,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C73 — C2×C73⋊C3 |
Generators and relations for C2×C73⋊C3
G = < a,b,c | a2=b73=c3=1, ab=ba, ac=ca, cbc-1=b64 >
Smallest permutation representation of C2×C73⋊C3
►On 146 points
Generators in S146
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 85)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 97)(25 98)(26 99)(27 100)(28 101)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 109)(37 110)(38 111)(39 112)(40 113)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 121)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 131)(59 132)(60 133)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 141)(69 142)(70 143)(71 144)(72 145)(73 146)
(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)
(2 9 65)(3 17 56)(4 25 47)(5 33 38)(6 41 29)(7 49 20)(8 57 11)(10 73 66)(12 16 48)(13 24 39)(14 32 30)(15 40 21)(18 64 67)(19 72 58)(22 23 31)(26 55 68)(27 63 59)(28 71 50)(34 46 69)(35 54 60)(36 62 51)(37 70 42)(43 45 61)(44 53 52)(75 82 138)(76 90 129)(77 98 120)(78 106 111)(79 114 102)(80 122 93)(81 130 84)(83 146 139)(85 89 121)(86 97 112)(87 105 103)(88 113 94)(91 137 140)(92 145 131)(95 96 104)(99 128 141)(100 136 132)(101 144 123)(107 119 142)(108 127 133)(109 135 124)(110 143 115)(116 118 134)(117 126 125)
G:=sub<Sym(146)| (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,85)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,97)(25,98)(26,99)(27,100)(28,101)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,113)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,141)(69,142)(70,143)(71,144)(72,145)(73,146), (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), (2,9,65)(3,17,56)(4,25,47)(5,33,38)(6,41,29)(7,49,20)(8,57,11)(10,73,66)(12,16,48)(13,24,39)(14,32,30)(15,40,21)(18,64,67)(19,72,58)(22,23,31)(26,55,68)(27,63,59)(28,71,50)(34,46,69)(35,54,60)(36,62,51)(37,70,42)(43,45,61)(44,53,52)(75,82,138)(76,90,129)(77,98,120)(78,106,111)(79,114,102)(80,122,93)(81,130,84)(83,146,139)(85,89,121)(86,97,112)(87,105,103)(88,113,94)(91,137,140)(92,145,131)(95,96,104)(99,128,141)(100,136,132)(101,144,123)(107,119,142)(108,127,133)(109,135,124)(110,143,115)(116,118,134)(117,126,125)>;
G:=Group( (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,85)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,97)(25,98)(26,99)(27,100)(28,101)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,113)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,141)(69,142)(70,143)(71,144)(72,145)(73,146), (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), (2,9,65)(3,17,56)(4,25,47)(5,33,38)(6,41,29)(7,49,20)(8,57,11)(10,73,66)(12,16,48)(13,24,39)(14,32,30)(15,40,21)(18,64,67)(19,72,58)(22,23,31)(26,55,68)(27,63,59)(28,71,50)(34,46,69)(35,54,60)(36,62,51)(37,70,42)(43,45,61)(44,53,52)(75,82,138)(76,90,129)(77,98,120)(78,106,111)(79,114,102)(80,122,93)(81,130,84)(83,146,139)(85,89,121)(86,97,112)(87,105,103)(88,113,94)(91,137,140)(92,145,131)(95,96,104)(99,128,141)(100,136,132)(101,144,123)(107,119,142)(108,127,133)(109,135,124)(110,143,115)(116,118,134)(117,126,125) );
G=PermutationGroup([[(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,81),(9,82),(10,83),(11,84),(12,85),(13,86),(14,87),(15,88),(16,89),(17,90),(18,91),(19,92),(20,93),(21,94),(22,95),(23,96),(24,97),(25,98),(26,99),(27,100),(28,101),(29,102),(30,103),(31,104),(32,105),(33,106),(34,107),(35,108),(36,109),(37,110),(38,111),(39,112),(40,113),(41,114),(42,115),(43,116),(44,117),(45,118),(46,119),(47,120),(48,121),(49,122),(50,123),(51,124),(52,125),(53,126),(54,127),(55,128),(56,129),(57,130),(58,131),(59,132),(60,133),(61,134),(62,135),(63,136),(64,137),(65,138),(66,139),(67,140),(68,141),(69,142),(70,143),(71,144),(72,145),(73,146)], [(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)], [(2,9,65),(3,17,56),(4,25,47),(5,33,38),(6,41,29),(7,49,20),(8,57,11),(10,73,66),(12,16,48),(13,24,39),(14,32,30),(15,40,21),(18,64,67),(19,72,58),(22,23,31),(26,55,68),(27,63,59),(28,71,50),(34,46,69),(35,54,60),(36,62,51),(37,70,42),(43,45,61),(44,53,52),(75,82,138),(76,90,129),(77,98,120),(78,106,111),(79,114,102),(80,122,93),(81,130,84),(83,146,139),(85,89,121),(86,97,112),(87,105,103),(88,113,94),(91,137,140),(92,145,131),(95,96,104),(99,128,141),(100,136,132),(101,144,123),(107,119,142),(108,127,133),(109,135,124),(110,143,115),(116,118,134),(117,126,125)]])
54 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 73A | ··· | 73X | 146A | ··· | 146X |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 73 | ··· | 73 | 146 | ··· | 146 |
| size | 1 | 1 | 73 | 73 | 73 | 73 | 3 | ··· | 3 | 3 | ··· | 3 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C2 | C3 | C6 | C73⋊C3 | C2×C73⋊C3 |
| kernel | C2×C73⋊C3 | C73⋊C3 | C146 | C73 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 24 | 24 |
Matrix representation of C2×C73⋊C3 ►in GL4(𝔽439) generated by
| 438 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 233 | 13 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 171 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 150 | 400 | 55 |
| 0 | 396 | 428 | 38 |
G:=sub<GL(4,GF(439))| [438,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,233,1,0,0,13,0,1,0,1,0,0],[171,0,0,0,0,1,150,396,0,0,400,428,0,0,55,38] >;
C2×C73⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_{73}\rtimes C_3 % in TeX
G:=Group("C2xC73:C3"); // GroupNames label
G:=SmallGroup(438,2);
// by ID
G=gap.SmallGroup(438,2);
# by ID
G:=PCGroup([3,-2,-3,-73,221]);
// Polycyclic
G:=Group<a,b,c|a^2=b^73=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^64>;
// generators/relations
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