metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊2D17, C136⋊2C2, C4.8D34, C34.1D4, C2.3D68, C17⋊1SD16, D68.1C2, Dic34⋊1C2, C68.8C22, SmallGroup(272,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C136⋊C2
G = < a,b | a136=b2=1, bab=a67 >
Smallest permutation representation of C136⋊C2
►On 136 points
Generators in S136
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136)
(2 68)(3 135)(4 66)(5 133)(6 64)(7 131)(8 62)(9 129)(10 60)(11 127)(12 58)(13 125)(14 56)(15 123)(16 54)(17 121)(18 52)(19 119)(20 50)(21 117)(22 48)(23 115)(24 46)(25 113)(26 44)(27 111)(28 42)(29 109)(30 40)(31 107)(32 38)(33 105)(34 36)(35 103)(37 101)(39 99)(41 97)(43 95)(45 93)(47 91)(49 89)(51 87)(53 85)(55 83)(57 81)(59 79)(61 77)(63 75)(65 73)(67 71)(70 136)(72 134)(74 132)(76 130)(78 128)(80 126)(82 124)(84 122)(86 120)(88 118)(90 116)(92 114)(94 112)(96 110)(98 108)(100 106)(102 104)
G:=sub<Sym(136)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (2,68)(3,135)(4,66)(5,133)(6,64)(7,131)(8,62)(9,129)(10,60)(11,127)(12,58)(13,125)(14,56)(15,123)(16,54)(17,121)(18,52)(19,119)(20,50)(21,117)(22,48)(23,115)(24,46)(25,113)(26,44)(27,111)(28,42)(29,109)(30,40)(31,107)(32,38)(33,105)(34,36)(35,103)(37,101)(39,99)(41,97)(43,95)(45,93)(47,91)(49,89)(51,87)(53,85)(55,83)(57,81)(59,79)(61,77)(63,75)(65,73)(67,71)(70,136)(72,134)(74,132)(76,130)(78,128)(80,126)(82,124)(84,122)(86,120)(88,118)(90,116)(92,114)(94,112)(96,110)(98,108)(100,106)(102,104)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (2,68)(3,135)(4,66)(5,133)(6,64)(7,131)(8,62)(9,129)(10,60)(11,127)(12,58)(13,125)(14,56)(15,123)(16,54)(17,121)(18,52)(19,119)(20,50)(21,117)(22,48)(23,115)(24,46)(25,113)(26,44)(27,111)(28,42)(29,109)(30,40)(31,107)(32,38)(33,105)(34,36)(35,103)(37,101)(39,99)(41,97)(43,95)(45,93)(47,91)(49,89)(51,87)(53,85)(55,83)(57,81)(59,79)(61,77)(63,75)(65,73)(67,71)(70,136)(72,134)(74,132)(76,130)(78,128)(80,126)(82,124)(84,122)(86,120)(88,118)(90,116)(92,114)(94,112)(96,110)(98,108)(100,106)(102,104) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)], [(2,68),(3,135),(4,66),(5,133),(6,64),(7,131),(8,62),(9,129),(10,60),(11,127),(12,58),(13,125),(14,56),(15,123),(16,54),(17,121),(18,52),(19,119),(20,50),(21,117),(22,48),(23,115),(24,46),(25,113),(26,44),(27,111),(28,42),(29,109),(30,40),(31,107),(32,38),(33,105),(34,36),(35,103),(37,101),(39,99),(41,97),(43,95),(45,93),(47,91),(49,89),(51,87),(53,85),(55,83),(57,81),(59,79),(61,77),(63,75),(65,73),(67,71),(70,136),(72,134),(74,132),(76,130),(78,128),(80,126),(82,124),(84,122),(86,120),(88,118),(90,116),(92,114),(94,112),(96,110),(98,108),(100,106),(102,104)]])
71 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 17A | ··· | 17H | 34A | ··· | 34H | 68A | ··· | 68P | 136A | ··· | 136AF |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 | 136 | ··· | 136 |
| size | 1 | 1 | 68 | 2 | 68 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
71 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | SD16 | D17 | D34 | D68 | C136⋊C2 |
| kernel | C136⋊C2 | C136 | Dic34 | D68 | C34 | C17 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 16 | 32 |
Matrix representation of C136⋊C2 ►in GL2(𝔽137) generated by
| 23 | 69 |
| 68 | 61 |
| 1 | 0 |
| 76 | 136 |
G:=sub<GL(2,GF(137))| [23,68,69,61],[1,76,0,136] >;
C136⋊C2 in GAP, Magma, Sage, TeX
C_{136}\rtimes C_2 % in TeX
G:=Group("C136:C2"); // GroupNames label
G:=SmallGroup(272,6);
// by ID
G=gap.SmallGroup(272,6);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,61,26,182,42,6404]);
// Polycyclic
G:=Group<a,b|a^136=b^2=1,b*a*b=a^67>;
// generators/relations
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