metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic38, C19⋊Q8, C4.D19, C76.1C2, C2.3D38, Dic19.C2, C38.1C22, SmallGroup(152,3)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic38
G = < a,b | a76=1, b2=a38, bab-1=a-1 >
Smallest permutation representation of Dic38
►Regular action 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)
(1 107 39 145)(2 106 40 144)(3 105 41 143)(4 104 42 142)(5 103 43 141)(6 102 44 140)(7 101 45 139)(8 100 46 138)(9 99 47 137)(10 98 48 136)(11 97 49 135)(12 96 50 134)(13 95 51 133)(14 94 52 132)(15 93 53 131)(16 92 54 130)(17 91 55 129)(18 90 56 128)(19 89 57 127)(20 88 58 126)(21 87 59 125)(22 86 60 124)(23 85 61 123)(24 84 62 122)(25 83 63 121)(26 82 64 120)(27 81 65 119)(28 80 66 118)(29 79 67 117)(30 78 68 116)(31 77 69 115)(32 152 70 114)(33 151 71 113)(34 150 72 112)(35 149 73 111)(36 148 74 110)(37 147 75 109)(38 146 76 108)
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), (1,107,39,145)(2,106,40,144)(3,105,41,143)(4,104,42,142)(5,103,43,141)(6,102,44,140)(7,101,45,139)(8,100,46,138)(9,99,47,137)(10,98,48,136)(11,97,49,135)(12,96,50,134)(13,95,51,133)(14,94,52,132)(15,93,53,131)(16,92,54,130)(17,91,55,129)(18,90,56,128)(19,89,57,127)(20,88,58,126)(21,87,59,125)(22,86,60,124)(23,85,61,123)(24,84,62,122)(25,83,63,121)(26,82,64,120)(27,81,65,119)(28,80,66,118)(29,79,67,117)(30,78,68,116)(31,77,69,115)(32,152,70,114)(33,151,71,113)(34,150,72,112)(35,149,73,111)(36,148,74,110)(37,147,75,109)(38,146,76,108)>;
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), (1,107,39,145)(2,106,40,144)(3,105,41,143)(4,104,42,142)(5,103,43,141)(6,102,44,140)(7,101,45,139)(8,100,46,138)(9,99,47,137)(10,98,48,136)(11,97,49,135)(12,96,50,134)(13,95,51,133)(14,94,52,132)(15,93,53,131)(16,92,54,130)(17,91,55,129)(18,90,56,128)(19,89,57,127)(20,88,58,126)(21,87,59,125)(22,86,60,124)(23,85,61,123)(24,84,62,122)(25,83,63,121)(26,82,64,120)(27,81,65,119)(28,80,66,118)(29,79,67,117)(30,78,68,116)(31,77,69,115)(32,152,70,114)(33,151,71,113)(34,150,72,112)(35,149,73,111)(36,148,74,110)(37,147,75,109)(38,146,76,108) );
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)], [(1,107,39,145),(2,106,40,144),(3,105,41,143),(4,104,42,142),(5,103,43,141),(6,102,44,140),(7,101,45,139),(8,100,46,138),(9,99,47,137),(10,98,48,136),(11,97,49,135),(12,96,50,134),(13,95,51,133),(14,94,52,132),(15,93,53,131),(16,92,54,130),(17,91,55,129),(18,90,56,128),(19,89,57,127),(20,88,58,126),(21,87,59,125),(22,86,60,124),(23,85,61,123),(24,84,62,122),(25,83,63,121),(26,82,64,120),(27,81,65,119),(28,80,66,118),(29,79,67,117),(30,78,68,116),(31,77,69,115),(32,152,70,114),(33,151,71,113),(34,150,72,112),(35,149,73,111),(36,148,74,110),(37,147,75,109),(38,146,76,108)]])
Dic38 is a maximal subgroup of
C152⋊C2 Dic76 D4.D19 C19⋊Q16 D76⋊5C2 D4⋊2D19 Q8×D19 Dic38⋊C3 C57⋊Q8 Dic114
Dic38 is a maximal quotient of
Dic19⋊C4 C76⋊C4 C57⋊Q8 Dic114
41 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 19A | ··· | 19I | 38A | ··· | 38I | 76A | ··· | 76R |
| order | 1 | 2 | 4 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 2 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
41 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D19 | D38 | Dic38 |
| kernel | Dic38 | Dic19 | C76 | C19 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 9 | 9 | 18 |
Matrix representation of Dic38 ►in GL2(𝔽229) generated by
| 41 | 157 |
| 8 | 53 |
| 34 | 156 |
| 88 | 195 |
G:=sub<GL(2,GF(229))| [41,8,157,53],[34,88,156,195] >;
Dic38 in GAP, Magma, Sage, TeX
{\rm Dic}_{38} % in TeX
G:=Group("Dic38"); // GroupNames label
G:=SmallGroup(152,3);
// by ID
G=gap.SmallGroup(152,3);
# by ID
G:=PCGroup([4,-2,-2,-2,-19,16,49,21,2307]);
// Polycyclic
G:=Group<a,b|a^76=1,b^2=a^38,b*a*b^-1=a^-1>;
// generators/relations
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