direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C2×C61⋊C3, C122⋊C3, C61⋊2C6, SmallGroup(366,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C61 — C2×C61⋊C3 |
Generators and relations for C2×C61⋊C3
G = < a,b,c | a2=b61=c3=1, ab=ba, ac=ca, cbc-1=b13 >
Smallest permutation representation of C2×C61⋊C3
►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)
(2 48 14)(3 34 27)(4 20 40)(5 6 53)(7 39 18)(8 25 31)(9 11 44)(10 58 57)(12 30 22)(13 16 35)(15 49 61)(17 21 26)(19 54 52)(23 59 43)(24 45 56)(28 50 47)(29 36 60)(32 55 38)(33 41 51)(37 46 42)(63 109 75)(64 95 88)(65 81 101)(66 67 114)(68 100 79)(69 86 92)(70 72 105)(71 119 118)(73 91 83)(74 77 96)(76 110 122)(78 82 87)(80 115 113)(84 120 104)(85 106 117)(89 111 108)(90 97 121)(93 116 99)(94 102 112)(98 107 103)
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), (2,48,14)(3,34,27)(4,20,40)(5,6,53)(7,39,18)(8,25,31)(9,11,44)(10,58,57)(12,30,22)(13,16,35)(15,49,61)(17,21,26)(19,54,52)(23,59,43)(24,45,56)(28,50,47)(29,36,60)(32,55,38)(33,41,51)(37,46,42)(63,109,75)(64,95,88)(65,81,101)(66,67,114)(68,100,79)(69,86,92)(70,72,105)(71,119,118)(73,91,83)(74,77,96)(76,110,122)(78,82,87)(80,115,113)(84,120,104)(85,106,117)(89,111,108)(90,97,121)(93,116,99)(94,102,112)(98,107,103)>;
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), (2,48,14)(3,34,27)(4,20,40)(5,6,53)(7,39,18)(8,25,31)(9,11,44)(10,58,57)(12,30,22)(13,16,35)(15,49,61)(17,21,26)(19,54,52)(23,59,43)(24,45,56)(28,50,47)(29,36,60)(32,55,38)(33,41,51)(37,46,42)(63,109,75)(64,95,88)(65,81,101)(66,67,114)(68,100,79)(69,86,92)(70,72,105)(71,119,118)(73,91,83)(74,77,96)(76,110,122)(78,82,87)(80,115,113)(84,120,104)(85,106,117)(89,111,108)(90,97,121)(93,116,99)(94,102,112)(98,107,103) );
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)], [(2,48,14),(3,34,27),(4,20,40),(5,6,53),(7,39,18),(8,25,31),(9,11,44),(10,58,57),(12,30,22),(13,16,35),(15,49,61),(17,21,26),(19,54,52),(23,59,43),(24,45,56),(28,50,47),(29,36,60),(32,55,38),(33,41,51),(37,46,42),(63,109,75),(64,95,88),(65,81,101),(66,67,114),(68,100,79),(69,86,92),(70,72,105),(71,119,118),(73,91,83),(74,77,96),(76,110,122),(78,82,87),(80,115,113),(84,120,104),(85,106,117),(89,111,108),(90,97,121),(93,116,99),(94,102,112),(98,107,103)]])
46 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 61A | ··· | 61T | 122A | ··· | 122T |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 61 | ··· | 61 | 122 | ··· | 122 |
| size | 1 | 1 | 61 | 61 | 61 | 61 | 3 | ··· | 3 | 3 | ··· | 3 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C2 | C3 | C6 | C61⋊C3 | C2×C61⋊C3 |
| kernel | C2×C61⋊C3 | C61⋊C3 | C122 | C61 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 20 | 20 |
Matrix representation of C2×C61⋊C3 ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 3 | 4 | 0 |
| 1 | 2 | 1 |
| 0 | 4 | 0 |
| 1 | 0 | 9 |
| 0 | 0 | 12 |
| 0 | 1 | 12 |
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,12],[3,1,0,4,2,4,0,1,0],[1,0,0,0,0,1,9,12,12] >;
C2×C61⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_{61}\rtimes C_3 % in TeX
G:=Group("C2xC61:C3"); // GroupNames label
G:=SmallGroup(366,2);
// by ID
G=gap.SmallGroup(366,2);
# by ID
G:=PCGroup([3,-2,-3,-61,1274]);
// Polycyclic
G:=Group<a,b,c|a^2=b^61=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^13>;
// generators/relations
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