direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary
Aliases: C3xC49:C3, C147:C3, C49:C32, C21.2(C7:C3), C7.(C3xC7:C3), SmallGroup(441,3)
Series: Derived ►Chief ►Lower central ►Upper central
C49 — C3xC49:C3 |
Generators and relations for C3xC49:C3
G = < a,b,c | a3=b49=c3=1, ab=ba, ac=ca, cbc-1=b18 >
(1 111 65)(2 112 66)(3 113 67)(4 114 68)(5 115 69)(6 116 70)(7 117 71)(8 118 72)(9 119 73)(10 120 74)(11 121 75)(12 122 76)(13 123 77)(14 124 78)(15 125 79)(16 126 80)(17 127 81)(18 128 82)(19 129 83)(20 130 84)(21 131 85)(22 132 86)(23 133 87)(24 134 88)(25 135 89)(26 136 90)(27 137 91)(28 138 92)(29 139 93)(30 140 94)(31 141 95)(32 142 96)(33 143 97)(34 144 98)(35 145 50)(36 146 51)(37 147 52)(38 99 53)(39 100 54)(40 101 55)(41 102 56)(42 103 57)(43 104 58)(44 105 59)(45 106 60)(46 107 61)(47 108 62)(48 109 63)(49 110 64)
(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)
(1 65 111)(2 95 129)(3 76 147)(4 57 116)(5 87 134)(6 68 103)(7 98 121)(8 79 139)(9 60 108)(10 90 126)(11 71 144)(12 52 113)(13 82 131)(14 63 100)(15 93 118)(16 74 136)(17 55 105)(18 85 123)(19 66 141)(20 96 110)(21 77 128)(22 58 146)(23 88 115)(24 69 133)(25 50 102)(26 80 120)(27 61 138)(28 91 107)(29 72 125)(30 53 143)(31 83 112)(32 64 130)(33 94 99)(34 75 117)(35 56 135)(36 86 104)(37 67 122)(38 97 140)(39 78 109)(40 59 127)(41 89 145)(42 70 114)(43 51 132)(44 81 101)(45 62 119)(46 92 137)(47 73 106)(48 54 124)(49 84 142)
G:=sub<Sym(147)| (1,111,65)(2,112,66)(3,113,67)(4,114,68)(5,115,69)(6,116,70)(7,117,71)(8,118,72)(9,119,73)(10,120,74)(11,121,75)(12,122,76)(13,123,77)(14,124,78)(15,125,79)(16,126,80)(17,127,81)(18,128,82)(19,129,83)(20,130,84)(21,131,85)(22,132,86)(23,133,87)(24,134,88)(25,135,89)(26,136,90)(27,137,91)(28,138,92)(29,139,93)(30,140,94)(31,141,95)(32,142,96)(33,143,97)(34,144,98)(35,145,50)(36,146,51)(37,147,52)(38,99,53)(39,100,54)(40,101,55)(41,102,56)(42,103,57)(43,104,58)(44,105,59)(45,106,60)(46,107,61)(47,108,62)(48,109,63)(49,110,64), (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), (1,65,111)(2,95,129)(3,76,147)(4,57,116)(5,87,134)(6,68,103)(7,98,121)(8,79,139)(9,60,108)(10,90,126)(11,71,144)(12,52,113)(13,82,131)(14,63,100)(15,93,118)(16,74,136)(17,55,105)(18,85,123)(19,66,141)(20,96,110)(21,77,128)(22,58,146)(23,88,115)(24,69,133)(25,50,102)(26,80,120)(27,61,138)(28,91,107)(29,72,125)(30,53,143)(31,83,112)(32,64,130)(33,94,99)(34,75,117)(35,56,135)(36,86,104)(37,67,122)(38,97,140)(39,78,109)(40,59,127)(41,89,145)(42,70,114)(43,51,132)(44,81,101)(45,62,119)(46,92,137)(47,73,106)(48,54,124)(49,84,142)>;
G:=Group( (1,111,65)(2,112,66)(3,113,67)(4,114,68)(5,115,69)(6,116,70)(7,117,71)(8,118,72)(9,119,73)(10,120,74)(11,121,75)(12,122,76)(13,123,77)(14,124,78)(15,125,79)(16,126,80)(17,127,81)(18,128,82)(19,129,83)(20,130,84)(21,131,85)(22,132,86)(23,133,87)(24,134,88)(25,135,89)(26,136,90)(27,137,91)(28,138,92)(29,139,93)(30,140,94)(31,141,95)(32,142,96)(33,143,97)(34,144,98)(35,145,50)(36,146,51)(37,147,52)(38,99,53)(39,100,54)(40,101,55)(41,102,56)(42,103,57)(43,104,58)(44,105,59)(45,106,60)(46,107,61)(47,108,62)(48,109,63)(49,110,64), (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), (1,65,111)(2,95,129)(3,76,147)(4,57,116)(5,87,134)(6,68,103)(7,98,121)(8,79,139)(9,60,108)(10,90,126)(11,71,144)(12,52,113)(13,82,131)(14,63,100)(15,93,118)(16,74,136)(17,55,105)(18,85,123)(19,66,141)(20,96,110)(21,77,128)(22,58,146)(23,88,115)(24,69,133)(25,50,102)(26,80,120)(27,61,138)(28,91,107)(29,72,125)(30,53,143)(31,83,112)(32,64,130)(33,94,99)(34,75,117)(35,56,135)(36,86,104)(37,67,122)(38,97,140)(39,78,109)(40,59,127)(41,89,145)(42,70,114)(43,51,132)(44,81,101)(45,62,119)(46,92,137)(47,73,106)(48,54,124)(49,84,142) );
G=PermutationGroup([[(1,111,65),(2,112,66),(3,113,67),(4,114,68),(5,115,69),(6,116,70),(7,117,71),(8,118,72),(9,119,73),(10,120,74),(11,121,75),(12,122,76),(13,123,77),(14,124,78),(15,125,79),(16,126,80),(17,127,81),(18,128,82),(19,129,83),(20,130,84),(21,131,85),(22,132,86),(23,133,87),(24,134,88),(25,135,89),(26,136,90),(27,137,91),(28,138,92),(29,139,93),(30,140,94),(31,141,95),(32,142,96),(33,143,97),(34,144,98),(35,145,50),(36,146,51),(37,147,52),(38,99,53),(39,100,54),(40,101,55),(41,102,56),(42,103,57),(43,104,58),(44,105,59),(45,106,60),(46,107,61),(47,108,62),(48,109,63),(49,110,64)], [(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)], [(1,65,111),(2,95,129),(3,76,147),(4,57,116),(5,87,134),(6,68,103),(7,98,121),(8,79,139),(9,60,108),(10,90,126),(11,71,144),(12,52,113),(13,82,131),(14,63,100),(15,93,118),(16,74,136),(17,55,105),(18,85,123),(19,66,141),(20,96,110),(21,77,128),(22,58,146),(23,88,115),(24,69,133),(25,50,102),(26,80,120),(27,61,138),(28,91,107),(29,72,125),(30,53,143),(31,83,112),(32,64,130),(33,94,99),(34,75,117),(35,56,135),(36,86,104),(37,67,122),(38,97,140),(39,78,109),(40,59,127),(41,89,145),(42,70,114),(43,51,132),(44,81,101),(45,62,119),(46,92,137),(47,73,106),(48,54,124),(49,84,142)]])
57 conjugacy classes
class | 1 | 3A | 3B | 3C | ··· | 3H | 7A | 7B | 21A | 21B | 21C | 21D | 49A | ··· | 49N | 147A | ··· | 147AB |
order | 1 | 3 | 3 | 3 | ··· | 3 | 7 | 7 | 21 | 21 | 21 | 21 | 49 | ··· | 49 | 147 | ··· | 147 |
size | 1 | 1 | 1 | 49 | ··· | 49 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
57 irreducible representations
dim | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
type | + | ||||||
image | C1 | C3 | C3 | C7:C3 | C3xC7:C3 | C49:C3 | C3xC49:C3 |
kernel | C3xC49:C3 | C49:C3 | C147 | C21 | C7 | C3 | C1 |
# reps | 1 | 6 | 2 | 2 | 4 | 14 | 28 |
Matrix representation of C3xC49:C3 ►in GL4(F883) generated by
545 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 |
0 | 344 | 567 | 877 |
0 | 722 | 22 | 155 |
0 | 2 | 561 | 497 |
1 | 0 | 0 | 0 |
0 | 522 | 1 | 362 |
0 | 1 | 0 | 0 |
0 | 1 | 1 | 361 |
G:=sub<GL(4,GF(883))| [545,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,344,722,2,0,567,22,561,0,877,155,497],[1,0,0,0,0,522,1,1,0,1,0,1,0,362,0,361] >;
C3xC49:C3 in GAP, Magma, Sage, TeX
C_3\times C_{49}\rtimes C_3
% in TeX
G:=Group("C3xC49:C3");
// GroupNames label
G:=SmallGroup(441,3);
// by ID
G=gap.SmallGroup(441,3);
# by ID
G:=PCGroup([4,-3,-3,-7,-7,974,178,2019]);
// Polycyclic
G:=Group<a,b,c|a^3=b^49=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^18>;
// generators/relations
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