direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×D41, C41⋊1D6, C3⋊1D82, D123⋊C2, C123⋊C22, (S3×C41)⋊C2, (C3×D41)⋊C2, SmallGroup(492,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C123 — S3×D41 |
Generators and relations for S3×D41
G = < a,b,c,d | a3=b2=c41=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of S3×D41
►On 123 points
Generators in S123
(1 53 123)(2 54 83)(3 55 84)(4 56 85)(5 57 86)(6 58 87)(7 59 88)(8 60 89)(9 61 90)(10 62 91)(11 63 92)(12 64 93)(13 65 94)(14 66 95)(15 67 96)(16 68 97)(17 69 98)(18 70 99)(19 71 100)(20 72 101)(21 73 102)(22 74 103)(23 75 104)(24 76 105)(25 77 106)(26 78 107)(27 79 108)(28 80 109)(29 81 110)(30 82 111)(31 42 112)(32 43 113)(33 44 114)(34 45 115)(35 46 116)(36 47 117)(37 48 118)(38 49 119)(39 50 120)(40 51 121)(41 52 122)
(42 112)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 121)(52 122)(53 123)(54 83)(55 84)(56 85)(57 86)(58 87)(59 88)(60 89)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 99)(71 100)(72 101)(73 102)(74 103)(75 104)(76 105)(77 106)(78 107)(79 108)(80 109)(81 110)(82 111)
(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 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(83 121)(84 120)(85 119)(86 118)(87 117)(88 116)(89 115)(90 114)(91 113)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)(122 123)
G:=sub<Sym(123)| (1,53,123)(2,54,83)(3,55,84)(4,56,85)(5,57,86)(6,58,87)(7,59,88)(8,60,89)(9,61,90)(10,62,91)(11,63,92)(12,64,93)(13,65,94)(14,66,95)(15,67,96)(16,68,97)(17,69,98)(18,70,99)(19,71,100)(20,72,101)(21,73,102)(22,74,103)(23,75,104)(24,76,105)(25,77,106)(26,78,107)(27,79,108)(28,80,109)(29,81,110)(30,82,111)(31,42,112)(32,43,113)(33,44,114)(34,45,115)(35,46,116)(36,47,117)(37,48,118)(38,49,119)(39,50,120)(40,51,121)(41,52,122), (42,112)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,121)(52,122)(53,123)(54,83)(55,84)(56,85)(57,86)(58,87)(59,88)(60,89)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,97)(69,98)(70,99)(71,100)(72,101)(73,102)(74,103)(75,104)(76,105)(77,106)(78,107)(79,108)(80,109)(81,110)(82,111), (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)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(83,121)(84,120)(85,119)(86,118)(87,117)(88,116)(89,115)(90,114)(91,113)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103)(122,123)>;
G:=Group( (1,53,123)(2,54,83)(3,55,84)(4,56,85)(5,57,86)(6,58,87)(7,59,88)(8,60,89)(9,61,90)(10,62,91)(11,63,92)(12,64,93)(13,65,94)(14,66,95)(15,67,96)(16,68,97)(17,69,98)(18,70,99)(19,71,100)(20,72,101)(21,73,102)(22,74,103)(23,75,104)(24,76,105)(25,77,106)(26,78,107)(27,79,108)(28,80,109)(29,81,110)(30,82,111)(31,42,112)(32,43,113)(33,44,114)(34,45,115)(35,46,116)(36,47,117)(37,48,118)(38,49,119)(39,50,120)(40,51,121)(41,52,122), (42,112)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,121)(52,122)(53,123)(54,83)(55,84)(56,85)(57,86)(58,87)(59,88)(60,89)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,97)(69,98)(70,99)(71,100)(72,101)(73,102)(74,103)(75,104)(76,105)(77,106)(78,107)(79,108)(80,109)(81,110)(82,111), (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)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(83,121)(84,120)(85,119)(86,118)(87,117)(88,116)(89,115)(90,114)(91,113)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103)(122,123) );
G=PermutationGroup([[(1,53,123),(2,54,83),(3,55,84),(4,56,85),(5,57,86),(6,58,87),(7,59,88),(8,60,89),(9,61,90),(10,62,91),(11,63,92),(12,64,93),(13,65,94),(14,66,95),(15,67,96),(16,68,97),(17,69,98),(18,70,99),(19,71,100),(20,72,101),(21,73,102),(22,74,103),(23,75,104),(24,76,105),(25,77,106),(26,78,107),(27,79,108),(28,80,109),(29,81,110),(30,82,111),(31,42,112),(32,43,113),(33,44,114),(34,45,115),(35,46,116),(36,47,117),(37,48,118),(38,49,119),(39,50,120),(40,51,121),(41,52,122)], [(42,112),(43,113),(44,114),(45,115),(46,116),(47,117),(48,118),(49,119),(50,120),(51,121),(52,122),(53,123),(54,83),(55,84),(56,85),(57,86),(58,87),(59,88),(60,89),(61,90),(62,91),(63,92),(64,93),(65,94),(66,95),(67,96),(68,97),(69,98),(70,99),(71,100),(72,101),(73,102),(74,103),(75,104),(76,105),(77,106),(78,107),(79,108),(80,109),(81,110),(82,111)], [(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),(42,63),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57),(49,56),(50,55),(51,54),(52,53),(64,82),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(83,121),(84,120),(85,119),(86,118),(87,117),(88,116),(89,115),(90,114),(91,113),(92,112),(93,111),(94,110),(95,109),(96,108),(97,107),(98,106),(99,105),(100,104),(101,103),(122,123)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 41A | ··· | 41T | 82A | ··· | 82T | 123A | ··· | 123T |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 41 | ··· | 41 | 82 | ··· | 82 | 123 | ··· | 123 |
| size | 1 | 3 | 41 | 123 | 2 | 82 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D6 | D41 | D82 | S3×D41 |
| kernel | S3×D41 | S3×C41 | C3×D41 | D123 | D41 | C41 | S3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 20 | 20 | 20 |
Matrix representation of S3×D41 ►in GL4(𝔽739) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 63 |
| 0 | 0 | 563 | 737 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 738 | 676 |
| 0 | 0 | 0 | 1 |
| 258 | 1 | 0 | 0 |
| 346 | 262 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 128 | 734 | 0 | 0 |
| 25 | 611 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(739))| [1,0,0,0,0,1,0,0,0,0,1,563,0,0,63,737],[1,0,0,0,0,1,0,0,0,0,738,0,0,0,676,1],[258,346,0,0,1,262,0,0,0,0,1,0,0,0,0,1],[128,25,0,0,734,611,0,0,0,0,1,0,0,0,0,1] >;
S3×D41 in GAP, Magma, Sage, TeX
S_3\times D_{41} % in TeX
G:=Group("S3xD41"); // GroupNames label
G:=SmallGroup(492,7);
// by ID
G=gap.SmallGroup(492,7);
# by ID
G:=PCGroup([4,-2,-2,-3,-41,54,7683]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^41=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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