direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary
Aliases: C3×C43⋊C3, C129⋊C3, C43⋊C32, SmallGroup(387,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C43 — C3×C43⋊C3 |
Generators and relations for C3×C43⋊C3
G = < a,b,c | a3=b43=c3=1, ab=ba, ac=ca, cbc-1=b6 >
Smallest permutation representation of C3×C43⋊C3
►On 129 points
Generators in S129
(1 87 44)(2 88 45)(3 89 46)(4 90 47)(5 91 48)(6 92 49)(7 93 50)(8 94 51)(9 95 52)(10 96 53)(11 97 54)(12 98 55)(13 99 56)(14 100 57)(15 101 58)(16 102 59)(17 103 60)(18 104 61)(19 105 62)(20 106 63)(21 107 64)(22 108 65)(23 109 66)(24 110 67)(25 111 68)(26 112 69)(27 113 70)(28 114 71)(29 115 72)(30 116 73)(31 117 74)(32 118 75)(33 119 76)(34 120 77)(35 121 78)(36 122 79)(37 123 80)(38 124 81)(39 125 82)(40 126 83)(41 127 84)(42 128 85)(43 129 86)
(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)
(1 44 87)(2 80 93)(3 73 99)(4 66 105)(5 59 111)(6 52 117)(7 45 123)(8 81 129)(9 74 92)(10 67 98)(11 60 104)(12 53 110)(13 46 116)(14 82 122)(15 75 128)(16 68 91)(17 61 97)(18 54 103)(19 47 109)(20 83 115)(21 76 121)(22 69 127)(23 62 90)(24 55 96)(25 48 102)(26 84 108)(27 77 114)(28 70 120)(29 63 126)(30 56 89)(31 49 95)(32 85 101)(33 78 107)(34 71 113)(35 64 119)(36 57 125)(37 50 88)(38 86 94)(39 79 100)(40 72 106)(41 65 112)(42 58 118)(43 51 124)
G:=sub<Sym(129)| (1,87,44)(2,88,45)(3,89,46)(4,90,47)(5,91,48)(6,92,49)(7,93,50)(8,94,51)(9,95,52)(10,96,53)(11,97,54)(12,98,55)(13,99,56)(14,100,57)(15,101,58)(16,102,59)(17,103,60)(18,104,61)(19,105,62)(20,106,63)(21,107,64)(22,108,65)(23,109,66)(24,110,67)(25,111,68)(26,112,69)(27,113,70)(28,114,71)(29,115,72)(30,116,73)(31,117,74)(32,118,75)(33,119,76)(34,120,77)(35,121,78)(36,122,79)(37,123,80)(38,124,81)(39,125,82)(40,126,83)(41,127,84)(42,128,85)(43,129,86), (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), (1,44,87)(2,80,93)(3,73,99)(4,66,105)(5,59,111)(6,52,117)(7,45,123)(8,81,129)(9,74,92)(10,67,98)(11,60,104)(12,53,110)(13,46,116)(14,82,122)(15,75,128)(16,68,91)(17,61,97)(18,54,103)(19,47,109)(20,83,115)(21,76,121)(22,69,127)(23,62,90)(24,55,96)(25,48,102)(26,84,108)(27,77,114)(28,70,120)(29,63,126)(30,56,89)(31,49,95)(32,85,101)(33,78,107)(34,71,113)(35,64,119)(36,57,125)(37,50,88)(38,86,94)(39,79,100)(40,72,106)(41,65,112)(42,58,118)(43,51,124)>;
G:=Group( (1,87,44)(2,88,45)(3,89,46)(4,90,47)(5,91,48)(6,92,49)(7,93,50)(8,94,51)(9,95,52)(10,96,53)(11,97,54)(12,98,55)(13,99,56)(14,100,57)(15,101,58)(16,102,59)(17,103,60)(18,104,61)(19,105,62)(20,106,63)(21,107,64)(22,108,65)(23,109,66)(24,110,67)(25,111,68)(26,112,69)(27,113,70)(28,114,71)(29,115,72)(30,116,73)(31,117,74)(32,118,75)(33,119,76)(34,120,77)(35,121,78)(36,122,79)(37,123,80)(38,124,81)(39,125,82)(40,126,83)(41,127,84)(42,128,85)(43,129,86), (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), (1,44,87)(2,80,93)(3,73,99)(4,66,105)(5,59,111)(6,52,117)(7,45,123)(8,81,129)(9,74,92)(10,67,98)(11,60,104)(12,53,110)(13,46,116)(14,82,122)(15,75,128)(16,68,91)(17,61,97)(18,54,103)(19,47,109)(20,83,115)(21,76,121)(22,69,127)(23,62,90)(24,55,96)(25,48,102)(26,84,108)(27,77,114)(28,70,120)(29,63,126)(30,56,89)(31,49,95)(32,85,101)(33,78,107)(34,71,113)(35,64,119)(36,57,125)(37,50,88)(38,86,94)(39,79,100)(40,72,106)(41,65,112)(42,58,118)(43,51,124) );
G=PermutationGroup([[(1,87,44),(2,88,45),(3,89,46),(4,90,47),(5,91,48),(6,92,49),(7,93,50),(8,94,51),(9,95,52),(10,96,53),(11,97,54),(12,98,55),(13,99,56),(14,100,57),(15,101,58),(16,102,59),(17,103,60),(18,104,61),(19,105,62),(20,106,63),(21,107,64),(22,108,65),(23,109,66),(24,110,67),(25,111,68),(26,112,69),(27,113,70),(28,114,71),(29,115,72),(30,116,73),(31,117,74),(32,118,75),(33,119,76),(34,120,77),(35,121,78),(36,122,79),(37,123,80),(38,124,81),(39,125,82),(40,126,83),(41,127,84),(42,128,85),(43,129,86)], [(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)], [(1,44,87),(2,80,93),(3,73,99),(4,66,105),(5,59,111),(6,52,117),(7,45,123),(8,81,129),(9,74,92),(10,67,98),(11,60,104),(12,53,110),(13,46,116),(14,82,122),(15,75,128),(16,68,91),(17,61,97),(18,54,103),(19,47,109),(20,83,115),(21,76,121),(22,69,127),(23,62,90),(24,55,96),(25,48,102),(26,84,108),(27,77,114),(28,70,120),(29,63,126),(30,56,89),(31,49,95),(32,85,101),(33,78,107),(34,71,113),(35,64,119),(36,57,125),(37,50,88),(38,86,94),(39,79,100),(40,72,106),(41,65,112),(42,58,118),(43,51,124)]])
51 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3H | 43A | ··· | 43N | 129A | ··· | 129AB |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 43 | ··· | 43 | 129 | ··· | 129 |
| size | 1 | 1 | 1 | 43 | ··· | 43 | 3 | ··· | 3 | 3 | ··· | 3 |
51 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 |
| type | + | ||||
| image | C1 | C3 | C3 | C43⋊C3 | C3×C43⋊C3 |
| kernel | C3×C43⋊C3 | C43⋊C3 | C129 | C3 | C1 |
| # reps | 1 | 6 | 2 | 14 | 28 |
Matrix representation of C3×C43⋊C3 ►in GL3(𝔽1033) generated by
| 837 | 0 | 0 |
| 0 | 837 | 0 |
| 0 | 0 | 837 |
| 0 | 0 | 1 |
| 1 | 0 | 831 |
| 0 | 1 | 476 |
| 195 | 702 | 259 |
| 0 | 471 | 385 |
| 0 | 98 | 367 |
G:=sub<GL(3,GF(1033))| [837,0,0,0,837,0,0,0,837],[0,1,0,0,0,1,1,831,476],[195,0,0,702,471,98,259,385,367] >;
C3×C43⋊C3 in GAP, Magma, Sage, TeX
C_3\times C_{43}\rtimes C_3 % in TeX
G:=Group("C3xC43:C3"); // GroupNames label
G:=SmallGroup(387,3);
// by ID
G=gap.SmallGroup(387,3);
# by ID
G:=PCGroup([3,-3,-3,-43,2918]);
// Polycyclic
G:=Group<a,b,c|a^3=b^43=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
// generators/relations
Export