metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D76⋊4C2, Q8⋊2D19, C4.7D38, C38.8C23, C76.7C22, D38.3C22, Dic19.5C22, (C4×D19)⋊3C2, C19⋊3(C4○D4), (Q8×C19)⋊3C2, C2.9(C22×D19), SmallGroup(304,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D76⋊C2
G = < a,b,c | a76=b2=c2=1, bab=a-1, cac=a37, cbc=a74b >
Smallest permutation representation of D76⋊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)
(1 76)(2 75)(3 74)(4 73)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(77 122)(78 121)(79 120)(80 119)(81 118)(82 117)(83 116)(84 115)(85 114)(86 113)(87 112)(88 111)(89 110)(90 109)(91 108)(92 107)(93 106)(94 105)(95 104)(96 103)(97 102)(98 101)(99 100)(123 152)(124 151)(125 150)(126 149)(127 148)(128 147)(129 146)(130 145)(131 144)(132 143)(133 142)(134 141)(135 140)(136 139)(137 138)
(1 119)(2 80)(3 117)(4 78)(5 115)(6 152)(7 113)(8 150)(9 111)(10 148)(11 109)(12 146)(13 107)(14 144)(15 105)(16 142)(17 103)(18 140)(19 101)(20 138)(21 99)(22 136)(23 97)(24 134)(25 95)(26 132)(27 93)(28 130)(29 91)(30 128)(31 89)(32 126)(33 87)(34 124)(35 85)(36 122)(37 83)(38 120)(39 81)(40 118)(41 79)(42 116)(43 77)(44 114)(45 151)(46 112)(47 149)(48 110)(49 147)(50 108)(51 145)(52 106)(53 143)(54 104)(55 141)(56 102)(57 139)(58 100)(59 137)(60 98)(61 135)(62 96)(63 133)(64 94)(65 131)(66 92)(67 129)(68 90)(69 127)(70 88)(71 125)(72 86)(73 123)(74 84)(75 121)(76 82)
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), (1,76)(2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(77,122)(78,121)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142)(134,141)(135,140)(136,139)(137,138), (1,119)(2,80)(3,117)(4,78)(5,115)(6,152)(7,113)(8,150)(9,111)(10,148)(11,109)(12,146)(13,107)(14,144)(15,105)(16,142)(17,103)(18,140)(19,101)(20,138)(21,99)(22,136)(23,97)(24,134)(25,95)(26,132)(27,93)(28,130)(29,91)(30,128)(31,89)(32,126)(33,87)(34,124)(35,85)(36,122)(37,83)(38,120)(39,81)(40,118)(41,79)(42,116)(43,77)(44,114)(45,151)(46,112)(47,149)(48,110)(49,147)(50,108)(51,145)(52,106)(53,143)(54,104)(55,141)(56,102)(57,139)(58,100)(59,137)(60,98)(61,135)(62,96)(63,133)(64,94)(65,131)(66,92)(67,129)(68,90)(69,127)(70,88)(71,125)(72,86)(73,123)(74,84)(75,121)(76,82)>;
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), (1,76)(2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(77,122)(78,121)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142)(134,141)(135,140)(136,139)(137,138), (1,119)(2,80)(3,117)(4,78)(5,115)(6,152)(7,113)(8,150)(9,111)(10,148)(11,109)(12,146)(13,107)(14,144)(15,105)(16,142)(17,103)(18,140)(19,101)(20,138)(21,99)(22,136)(23,97)(24,134)(25,95)(26,132)(27,93)(28,130)(29,91)(30,128)(31,89)(32,126)(33,87)(34,124)(35,85)(36,122)(37,83)(38,120)(39,81)(40,118)(41,79)(42,116)(43,77)(44,114)(45,151)(46,112)(47,149)(48,110)(49,147)(50,108)(51,145)(52,106)(53,143)(54,104)(55,141)(56,102)(57,139)(58,100)(59,137)(60,98)(61,135)(62,96)(63,133)(64,94)(65,131)(66,92)(67,129)(68,90)(69,127)(70,88)(71,125)(72,86)(73,123)(74,84)(75,121)(76,82) );
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)], [(1,76),(2,75),(3,74),(4,73),(5,72),(6,71),(7,70),(8,69),(9,68),(10,67),(11,66),(12,65),(13,64),(14,63),(15,62),(16,61),(17,60),(18,59),(19,58),(20,57),(21,56),(22,55),(23,54),(24,53),(25,52),(26,51),(27,50),(28,49),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39),(77,122),(78,121),(79,120),(80,119),(81,118),(82,117),(83,116),(84,115),(85,114),(86,113),(87,112),(88,111),(89,110),(90,109),(91,108),(92,107),(93,106),(94,105),(95,104),(96,103),(97,102),(98,101),(99,100),(123,152),(124,151),(125,150),(126,149),(127,148),(128,147),(129,146),(130,145),(131,144),(132,143),(133,142),(134,141),(135,140),(136,139),(137,138)], [(1,119),(2,80),(3,117),(4,78),(5,115),(6,152),(7,113),(8,150),(9,111),(10,148),(11,109),(12,146),(13,107),(14,144),(15,105),(16,142),(17,103),(18,140),(19,101),(20,138),(21,99),(22,136),(23,97),(24,134),(25,95),(26,132),(27,93),(28,130),(29,91),(30,128),(31,89),(32,126),(33,87),(34,124),(35,85),(36,122),(37,83),(38,120),(39,81),(40,118),(41,79),(42,116),(43,77),(44,114),(45,151),(46,112),(47,149),(48,110),(49,147),(50,108),(51,145),(52,106),(53,143),(54,104),(55,141),(56,102),(57,139),(58,100),(59,137),(60,98),(61,135),(62,96),(63,133),(64,94),(65,131),(66,92),(67,129),(68,90),(69,127),(70,88),(71,125),(72,86),(73,123),(74,84),(75,121),(76,82)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 19A | ··· | 19I | 38A | ··· | 38I | 76A | ··· | 76AA |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 38 | 38 | 38 | 2 | 2 | 2 | 19 | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C4○D4 | D19 | D38 | D76⋊C2 |
| kernel | D76⋊C2 | C4×D19 | D76 | Q8×C19 | C19 | Q8 | C4 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 9 | 27 | 9 |
Matrix representation of D76⋊C2 ►in GL4(𝔽229) generated by
| 187 | 117 | 0 | 0 |
| 112 | 86 | 0 | 0 |
| 0 | 0 | 228 | 22 |
| 0 | 0 | 104 | 1 |
| 187 | 117 | 0 | 0 |
| 165 | 42 | 0 | 0 |
| 0 | 0 | 228 | 22 |
| 0 | 0 | 0 | 1 |
| 228 | 0 | 0 | 0 |
| 132 | 1 | 0 | 0 |
| 0 | 0 | 107 | 165 |
| 0 | 0 | 93 | 122 |
G:=sub<GL(4,GF(229))| [187,112,0,0,117,86,0,0,0,0,228,104,0,0,22,1],[187,165,0,0,117,42,0,0,0,0,228,0,0,0,22,1],[228,132,0,0,0,1,0,0,0,0,107,93,0,0,165,122] >;
D76⋊C2 in GAP, Magma, Sage, TeX
D_{76}\rtimes C_2 % in TeX
G:=Group("D76:C2"); // GroupNames label
G:=SmallGroup(304,34);
// by ID
G=gap.SmallGroup(304,34);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,46,182,97,42,7204]);
// Polycyclic
G:=Group<a,b,c|a^76=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^37,c*b*c=a^74*b>;
// generators/relations
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