metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊2D19, C152⋊2C2, C38.1D4, C4.8D38, C2.3D76, C19⋊1SD16, D76.1C2, Dic38⋊1C2, C76.8C22, SmallGroup(304,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C152⋊C2
G = < a,b | a152=b2=1, bab=a75 >
Smallest permutation representation of C152⋊C2
►On 152 points
Generators in S152
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152)
(2 76)(3 151)(4 74)(5 149)(6 72)(7 147)(8 70)(9 145)(10 68)(11 143)(12 66)(13 141)(14 64)(15 139)(16 62)(17 137)(18 60)(19 135)(20 58)(21 133)(22 56)(23 131)(24 54)(25 129)(26 52)(27 127)(28 50)(29 125)(30 48)(31 123)(32 46)(33 121)(34 44)(35 119)(36 42)(37 117)(38 40)(39 115)(41 113)(43 111)(45 109)(47 107)(49 105)(51 103)(53 101)(55 99)(57 97)(59 95)(61 93)(63 91)(65 89)(67 87)(69 85)(71 83)(73 81)(75 79)(78 152)(80 150)(82 148)(84 146)(86 144)(88 142)(90 140)(92 138)(94 136)(96 134)(98 132)(100 130)(102 128)(104 126)(106 124)(108 122)(110 120)(112 118)(114 116)
G:=sub<Sym(152)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (2,76)(3,151)(4,74)(5,149)(6,72)(7,147)(8,70)(9,145)(10,68)(11,143)(12,66)(13,141)(14,64)(15,139)(16,62)(17,137)(18,60)(19,135)(20,58)(21,133)(22,56)(23,131)(24,54)(25,129)(26,52)(27,127)(28,50)(29,125)(30,48)(31,123)(32,46)(33,121)(34,44)(35,119)(36,42)(37,117)(38,40)(39,115)(41,113)(43,111)(45,109)(47,107)(49,105)(51,103)(53,101)(55,99)(57,97)(59,95)(61,93)(63,91)(65,89)(67,87)(69,85)(71,83)(73,81)(75,79)(78,152)(80,150)(82,148)(84,146)(86,144)(88,142)(90,140)(92,138)(94,136)(96,134)(98,132)(100,130)(102,128)(104,126)(106,124)(108,122)(110,120)(112,118)(114,116)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (2,76)(3,151)(4,74)(5,149)(6,72)(7,147)(8,70)(9,145)(10,68)(11,143)(12,66)(13,141)(14,64)(15,139)(16,62)(17,137)(18,60)(19,135)(20,58)(21,133)(22,56)(23,131)(24,54)(25,129)(26,52)(27,127)(28,50)(29,125)(30,48)(31,123)(32,46)(33,121)(34,44)(35,119)(36,42)(37,117)(38,40)(39,115)(41,113)(43,111)(45,109)(47,107)(49,105)(51,103)(53,101)(55,99)(57,97)(59,95)(61,93)(63,91)(65,89)(67,87)(69,85)(71,83)(73,81)(75,79)(78,152)(80,150)(82,148)(84,146)(86,144)(88,142)(90,140)(92,138)(94,136)(96,134)(98,132)(100,130)(102,128)(104,126)(106,124)(108,122)(110,120)(112,118)(114,116) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)], [(2,76),(3,151),(4,74),(5,149),(6,72),(7,147),(8,70),(9,145),(10,68),(11,143),(12,66),(13,141),(14,64),(15,139),(16,62),(17,137),(18,60),(19,135),(20,58),(21,133),(22,56),(23,131),(24,54),(25,129),(26,52),(27,127),(28,50),(29,125),(30,48),(31,123),(32,46),(33,121),(34,44),(35,119),(36,42),(37,117),(38,40),(39,115),(41,113),(43,111),(45,109),(47,107),(49,105),(51,103),(53,101),(55,99),(57,97),(59,95),(61,93),(63,91),(65,89),(67,87),(69,85),(71,83),(73,81),(75,79),(78,152),(80,150),(82,148),(84,146),(86,144),(88,142),(90,140),(92,138),(94,136),(96,134),(98,132),(100,130),(102,128),(104,126),(106,124),(108,122),(110,120),(112,118),(114,116)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 19A | ··· | 19I | 38A | ··· | 38I | 76A | ··· | 76R | 152A | ··· | 152AJ |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 | 152 | ··· | 152 |
| size | 1 | 1 | 76 | 2 | 76 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | SD16 | D19 | D38 | D76 | C152⋊C2 |
| kernel | C152⋊C2 | C152 | Dic38 | D76 | C38 | C19 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 9 | 9 | 18 | 36 |
Matrix representation of C152⋊C2 ►in GL2(𝔽457) generated by
| 385 | 101 |
| 307 | 166 |
| 370 | 100 |
| 34 | 87 |
G:=sub<GL(2,GF(457))| [385,307,101,166],[370,34,100,87] >;
C152⋊C2 in GAP, Magma, Sage, TeX
C_{152}\rtimes C_2 % in TeX
G:=Group("C152:C2"); // GroupNames label
G:=SmallGroup(304,5);
// by ID
G=gap.SmallGroup(304,5);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,61,26,182,42,7204]);
// Polycyclic
G:=Group<a,b|a^152=b^2=1,b*a*b=a^75>;
// generators/relations
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