direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×D23, C46⋊C6, C138⋊2C2, C69⋊3C22, C23⋊(C2×C6), SmallGroup(276,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — C6×D23 |
Generators and relations for C6×D23
G = < a,b,c | a6=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C6×D23
►On 138 points
Generators in S138
(1 99 50 76 36 116)(2 100 51 77 37 117)(3 101 52 78 38 118)(4 102 53 79 39 119)(5 103 54 80 40 120)(6 104 55 81 41 121)(7 105 56 82 42 122)(8 106 57 83 43 123)(9 107 58 84 44 124)(10 108 59 85 45 125)(11 109 60 86 46 126)(12 110 61 87 24 127)(13 111 62 88 25 128)(14 112 63 89 26 129)(15 113 64 90 27 130)(16 114 65 91 28 131)(17 115 66 92 29 132)(18 93 67 70 30 133)(19 94 68 71 31 134)(20 95 69 72 32 135)(21 96 47 73 33 136)(22 97 48 74 34 137)(23 98 49 75 35 138)
(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)
(1 75)(2 74)(3 73)(4 72)(5 71)(6 70)(7 92)(8 91)(9 90)(10 89)(11 88)(12 87)(13 86)(14 85)(15 84)(16 83)(17 82)(18 81)(19 80)(20 79)(21 78)(22 77)(23 76)(24 110)(25 109)(26 108)(27 107)(28 106)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 99)(36 98)(37 97)(38 96)(39 95)(40 94)(41 93)(42 115)(43 114)(44 113)(45 112)(46 111)(47 118)(48 117)(49 116)(50 138)(51 137)(52 136)(53 135)(54 134)(55 133)(56 132)(57 131)(58 130)(59 129)(60 128)(61 127)(62 126)(63 125)(64 124)(65 123)(66 122)(67 121)(68 120)(69 119)
G:=sub<Sym(138)| (1,99,50,76,36,116)(2,100,51,77,37,117)(3,101,52,78,38,118)(4,102,53,79,39,119)(5,103,54,80,40,120)(6,104,55,81,41,121)(7,105,56,82,42,122)(8,106,57,83,43,123)(9,107,58,84,44,124)(10,108,59,85,45,125)(11,109,60,86,46,126)(12,110,61,87,24,127)(13,111,62,88,25,128)(14,112,63,89,26,129)(15,113,64,90,27,130)(16,114,65,91,28,131)(17,115,66,92,29,132)(18,93,67,70,30,133)(19,94,68,71,31,134)(20,95,69,72,32,135)(21,96,47,73,33,136)(22,97,48,74,34,137)(23,98,49,75,35,138), (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), (1,75)(2,74)(3,73)(4,72)(5,71)(6,70)(7,92)(8,91)(9,90)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,110)(25,109)(26,108)(27,107)(28,106)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,115)(43,114)(44,113)(45,112)(46,111)(47,118)(48,117)(49,116)(50,138)(51,137)(52,136)(53,135)(54,134)(55,133)(56,132)(57,131)(58,130)(59,129)(60,128)(61,127)(62,126)(63,125)(64,124)(65,123)(66,122)(67,121)(68,120)(69,119)>;
G:=Group( (1,99,50,76,36,116)(2,100,51,77,37,117)(3,101,52,78,38,118)(4,102,53,79,39,119)(5,103,54,80,40,120)(6,104,55,81,41,121)(7,105,56,82,42,122)(8,106,57,83,43,123)(9,107,58,84,44,124)(10,108,59,85,45,125)(11,109,60,86,46,126)(12,110,61,87,24,127)(13,111,62,88,25,128)(14,112,63,89,26,129)(15,113,64,90,27,130)(16,114,65,91,28,131)(17,115,66,92,29,132)(18,93,67,70,30,133)(19,94,68,71,31,134)(20,95,69,72,32,135)(21,96,47,73,33,136)(22,97,48,74,34,137)(23,98,49,75,35,138), (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), (1,75)(2,74)(3,73)(4,72)(5,71)(6,70)(7,92)(8,91)(9,90)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,110)(25,109)(26,108)(27,107)(28,106)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,115)(43,114)(44,113)(45,112)(46,111)(47,118)(48,117)(49,116)(50,138)(51,137)(52,136)(53,135)(54,134)(55,133)(56,132)(57,131)(58,130)(59,129)(60,128)(61,127)(62,126)(63,125)(64,124)(65,123)(66,122)(67,121)(68,120)(69,119) );
G=PermutationGroup([[(1,99,50,76,36,116),(2,100,51,77,37,117),(3,101,52,78,38,118),(4,102,53,79,39,119),(5,103,54,80,40,120),(6,104,55,81,41,121),(7,105,56,82,42,122),(8,106,57,83,43,123),(9,107,58,84,44,124),(10,108,59,85,45,125),(11,109,60,86,46,126),(12,110,61,87,24,127),(13,111,62,88,25,128),(14,112,63,89,26,129),(15,113,64,90,27,130),(16,114,65,91,28,131),(17,115,66,92,29,132),(18,93,67,70,30,133),(19,94,68,71,31,134),(20,95,69,72,32,135),(21,96,47,73,33,136),(22,97,48,74,34,137),(23,98,49,75,35,138)], [(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)], [(1,75),(2,74),(3,73),(4,72),(5,71),(6,70),(7,92),(8,91),(9,90),(10,89),(11,88),(12,87),(13,86),(14,85),(15,84),(16,83),(17,82),(18,81),(19,80),(20,79),(21,78),(22,77),(23,76),(24,110),(25,109),(26,108),(27,107),(28,106),(29,105),(30,104),(31,103),(32,102),(33,101),(34,100),(35,99),(36,98),(37,97),(38,96),(39,95),(40,94),(41,93),(42,115),(43,114),(44,113),(45,112),(46,111),(47,118),(48,117),(49,116),(50,138),(51,137),(52,136),(53,135),(54,134),(55,133),(56,132),(57,131),(58,130),(59,129),(60,128),(61,127),(62,126),(63,125),(64,124),(65,123),(66,122),(67,121),(68,120),(69,119)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 23A | ··· | 23K | 46A | ··· | 46K | 69A | ··· | 69V | 138A | ··· | 138V |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 23 | ··· | 23 | 46 | ··· | 46 | 69 | ··· | 69 | 138 | ··· | 138 |
| size | 1 | 1 | 23 | 23 | 1 | 1 | 1 | 1 | 23 | 23 | 23 | 23 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D23 | D46 | C3×D23 | C6×D23 |
| kernel | C6×D23 | C3×D23 | C138 | D46 | D23 | C46 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 11 | 11 | 22 | 22 |
Matrix representation of C6×D23 ►in GL2(𝔽139) generated by
| 97 | 0 |
| 0 | 97 |
| 0 | 1 |
| 138 | 44 |
| 0 | 138 |
| 138 | 0 |
G:=sub<GL(2,GF(139))| [97,0,0,97],[0,138,1,44],[0,138,138,0] >;
C6×D23 in GAP, Magma, Sage, TeX
C_6\times D_{23} % in TeX
G:=Group("C6xD23"); // GroupNames label
G:=SmallGroup(276,7);
// by ID
G=gap.SmallGroup(276,7);
# by ID
G:=PCGroup([4,-2,-2,-3,-23,4227]);
// Polycyclic
G:=Group<a,b,c|a^6=b^23=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export