direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C61⋊C4, C122⋊C4, D61⋊C4, D122.C2, D61.C22, C61⋊(C2×C4), SmallGroup(488,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C61 — C2×C61⋊C4 |
Generators and relations for C2×C61⋊C4
G = < a,b,c | a2=b61=c4=1, ab=ba, ac=ca, cbc-1=b50 >
Smallest permutation representation of C2×C61⋊C4
►On 122 points
Generators in S122
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 97)(37 98)(38 99)(39 100)(40 101)(41 102)(42 103)(43 104)(44 105)(45 106)(46 107)(47 108)(48 109)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 121)(61 122)
(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)
(1 62)(2 73 61 112)(3 84 60 101)(4 95 59 90)(5 106 58 79)(6 117 57 68)(7 67 56 118)(8 78 55 107)(9 89 54 96)(10 100 53 85)(11 111 52 74)(12 122 51 63)(13 72 50 113)(14 83 49 102)(15 94 48 91)(16 105 47 80)(17 116 46 69)(18 66 45 119)(19 77 44 108)(20 88 43 97)(21 99 42 86)(22 110 41 75)(23 121 40 64)(24 71 39 114)(25 82 38 103)(26 93 37 92)(27 104 36 81)(28 115 35 70)(29 65 34 120)(30 76 33 109)(31 87 32 98)
G:=sub<Sym(122)| (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,103)(43,104)(44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122), (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), (1,62)(2,73,61,112)(3,84,60,101)(4,95,59,90)(5,106,58,79)(6,117,57,68)(7,67,56,118)(8,78,55,107)(9,89,54,96)(10,100,53,85)(11,111,52,74)(12,122,51,63)(13,72,50,113)(14,83,49,102)(15,94,48,91)(16,105,47,80)(17,116,46,69)(18,66,45,119)(19,77,44,108)(20,88,43,97)(21,99,42,86)(22,110,41,75)(23,121,40,64)(24,71,39,114)(25,82,38,103)(26,93,37,92)(27,104,36,81)(28,115,35,70)(29,65,34,120)(30,76,33,109)(31,87,32,98)>;
G:=Group( (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,103)(43,104)(44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122), (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), (1,62)(2,73,61,112)(3,84,60,101)(4,95,59,90)(5,106,58,79)(6,117,57,68)(7,67,56,118)(8,78,55,107)(9,89,54,96)(10,100,53,85)(11,111,52,74)(12,122,51,63)(13,72,50,113)(14,83,49,102)(15,94,48,91)(16,105,47,80)(17,116,46,69)(18,66,45,119)(19,77,44,108)(20,88,43,97)(21,99,42,86)(22,110,41,75)(23,121,40,64)(24,71,39,114)(25,82,38,103)(26,93,37,92)(27,104,36,81)(28,115,35,70)(29,65,34,120)(30,76,33,109)(31,87,32,98) );
G=PermutationGroup([[(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,73),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,85),(25,86),(26,87),(27,88),(28,89),(29,90),(30,91),(31,92),(32,93),(33,94),(34,95),(35,96),(36,97),(37,98),(38,99),(39,100),(40,101),(41,102),(42,103),(43,104),(44,105),(45,106),(46,107),(47,108),(48,109),(49,110),(50,111),(51,112),(52,113),(53,114),(54,115),(55,116),(56,117),(57,118),(58,119),(59,120),(60,121),(61,122)], [(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)], [(1,62),(2,73,61,112),(3,84,60,101),(4,95,59,90),(5,106,58,79),(6,117,57,68),(7,67,56,118),(8,78,55,107),(9,89,54,96),(10,100,53,85),(11,111,52,74),(12,122,51,63),(13,72,50,113),(14,83,49,102),(15,94,48,91),(16,105,47,80),(17,116,46,69),(18,66,45,119),(19,77,44,108),(20,88,43,97),(21,99,42,86),(22,110,41,75),(23,121,40,64),(24,71,39,114),(25,82,38,103),(26,93,37,92),(27,104,36,81),(28,115,35,70),(29,65,34,120),(30,76,33,109),(31,87,32,98)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 61A | ··· | 61O | 122A | ··· | 122O |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 61 | ··· | 61 | 122 | ··· | 122 |
| size | 1 | 1 | 61 | 61 | 61 | 61 | 61 | 61 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C4 | C4 | C61⋊C4 | C2×C61⋊C4 |
| kernel | C2×C61⋊C4 | C61⋊C4 | D122 | D61 | C122 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 15 | 15 |
Matrix representation of C2×C61⋊C4 ►in GL4(𝔽733) generated by
| 732 | 0 | 0 | 0 |
| 0 | 732 | 0 | 0 |
| 0 | 0 | 732 | 0 |
| 0 | 0 | 0 | 732 |
| 136 | 226 | 136 | 732 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 732 | 0 | 0 | 0 |
| 726 | 502 | 618 | 625 |
| 124 | 369 | 530 | 108 |
| 479 | 452 | 393 | 435 |
G:=sub<GL(4,GF(733))| [732,0,0,0,0,732,0,0,0,0,732,0,0,0,0,732],[136,1,0,0,226,0,1,0,136,0,0,1,732,0,0,0],[732,726,124,479,0,502,369,452,0,618,530,393,0,625,108,435] >;
C2×C61⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_{61}\rtimes C_4 % in TeX
G:=Group("C2xC61:C4"); // GroupNames label
G:=SmallGroup(488,12);
// by ID
G=gap.SmallGroup(488,12);
# by ID
G:=PCGroup([4,-2,-2,-2,-61,16,1411,1931]);
// Polycyclic
G:=Group<a,b,c|a^2=b^61=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^50>;
// generators/relations
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