direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D41, C41⋊C6, C123⋊2C2, SmallGroup(246,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C41 — C3×D41 |
Generators and relations for C3×D41
G = < a,b,c | a3=b41=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D41
►On 123 points
Generators in S123
(1 84 55)(2 85 56)(3 86 57)(4 87 58)(5 88 59)(6 89 60)(7 90 61)(8 91 62)(9 92 63)(10 93 64)(11 94 65)(12 95 66)(13 96 67)(14 97 68)(15 98 69)(16 99 70)(17 100 71)(18 101 72)(19 102 73)(20 103 74)(21 104 75)(22 105 76)(23 106 77)(24 107 78)(25 108 79)(26 109 80)(27 110 81)(28 111 82)(29 112 42)(30 113 43)(31 114 44)(32 115 45)(33 116 46)(34 117 47)(35 118 48)(36 119 49)(37 120 50)(38 121 51)(39 122 52)(40 123 53)(41 83 54)
(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)
(1 41)(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(68 82)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)(83 84)(85 123)(86 122)(87 121)(88 120)(89 119)(90 118)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)
G:=sub<Sym(123)| (1,84,55)(2,85,56)(3,86,57)(4,87,58)(5,88,59)(6,89,60)(7,90,61)(8,91,62)(9,92,63)(10,93,64)(11,94,65)(12,95,66)(13,96,67)(14,97,68)(15,98,69)(16,99,70)(17,100,71)(18,101,72)(19,102,73)(20,103,74)(21,104,75)(22,105,76)(23,106,77)(24,107,78)(25,108,79)(26,109,80)(27,110,81)(28,111,82)(29,112,42)(30,113,43)(31,114,44)(32,115,45)(33,116,46)(34,117,47)(35,118,48)(36,119,49)(37,120,50)(38,121,51)(39,122,52)(40,123,53)(41,83,54), (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), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(83,84)(85,123)(86,122)(87,121)(88,120)(89,119)(90,118)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)>;
G:=Group( (1,84,55)(2,85,56)(3,86,57)(4,87,58)(5,88,59)(6,89,60)(7,90,61)(8,91,62)(9,92,63)(10,93,64)(11,94,65)(12,95,66)(13,96,67)(14,97,68)(15,98,69)(16,99,70)(17,100,71)(18,101,72)(19,102,73)(20,103,74)(21,104,75)(22,105,76)(23,106,77)(24,107,78)(25,108,79)(26,109,80)(27,110,81)(28,111,82)(29,112,42)(30,113,43)(31,114,44)(32,115,45)(33,116,46)(34,117,47)(35,118,48)(36,119,49)(37,120,50)(38,121,51)(39,122,52)(40,123,53)(41,83,54), (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), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(83,84)(85,123)(86,122)(87,121)(88,120)(89,119)(90,118)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105) );
G=PermutationGroup([[(1,84,55),(2,85,56),(3,86,57),(4,87,58),(5,88,59),(6,89,60),(7,90,61),(8,91,62),(9,92,63),(10,93,64),(11,94,65),(12,95,66),(13,96,67),(14,97,68),(15,98,69),(16,99,70),(17,100,71),(18,101,72),(19,102,73),(20,103,74),(21,104,75),(22,105,76),(23,106,77),(24,107,78),(25,108,79),(26,109,80),(27,110,81),(28,111,82),(29,112,42),(30,113,43),(31,114,44),(32,115,45),(33,116,46),(34,117,47),(35,118,48),(36,119,49),(37,120,50),(38,121,51),(39,122,52),(40,123,53),(41,83,54)], [(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)], [(1,41),(2,40),(3,39),(4,38),(5,37),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22),(42,67),(43,66),(44,65),(45,64),(46,63),(47,62),(48,61),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(68,82),(69,81),(70,80),(71,79),(72,78),(73,77),(74,76),(83,84),(85,123),(86,122),(87,121),(88,120),(89,119),(90,118),(91,117),(92,116),(93,115),(94,114),(95,113),(96,112),(97,111),(98,110),(99,109),(100,108),(101,107),(102,106),(103,105)]])
C3×D41 is a maximal subgroup of
C41⋊Dic3
66 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 41A | ··· | 41T | 123A | ··· | 123AN |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 41 | ··· | 41 | 123 | ··· | 123 |
| size | 1 | 41 | 1 | 1 | 41 | 41 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D41 | C3×D41 |
| kernel | C3×D41 | C123 | D41 | C41 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 20 | 40 |
Matrix representation of C3×D41 ►in GL3(𝔽739) generated by
| 320 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 716 | 1 |
| 0 | 54 | 94 |
| 738 | 0 | 0 |
| 0 | 717 | 71 |
| 0 | 243 | 22 |
G:=sub<GL(3,GF(739))| [320,0,0,0,1,0,0,0,1],[1,0,0,0,716,54,0,1,94],[738,0,0,0,717,243,0,71,22] >;
C3×D41 in GAP, Magma, Sage, TeX
C_3\times D_{41} % in TeX
G:=Group("C3xD41"); // GroupNames label
G:=SmallGroup(246,2);
// by ID
G=gap.SmallGroup(246,2);
# by ID
G:=PCGroup([3,-2,-3,-41,2162]);
// Polycyclic
G:=Group<a,b,c|a^3=b^41=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export