direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4oD4xC19, D4:2C38, Q8:2C38, C38.13C23, C76.21C22, (C2xC4):3C38, (C2xC76):7C2, (D4xC19):5C2, C4.5(C2xC38), (Q8xC19):5C2, C22.(C2xC38), (C2xC38).2C22, C2.3(C22xC38), SmallGroup(304,40)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4oD4xC19
G = < a,b,c,d | a19=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Quotients: C1, C2, C22, C23, C4oD4, C19, C38, C2xC38, C22xC38, C4oD4xC19
Smallest permutation representation of C4oD4xC19
►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 36 120 135)(2 37 121 136)(3 38 122 137)(4 20 123 138)(5 21 124 139)(6 22 125 140)(7 23 126 141)(8 24 127 142)(9 25 128 143)(10 26 129 144)(11 27 130 145)(12 28 131 146)(13 29 132 147)(14 30 133 148)(15 31 115 149)(16 32 116 150)(17 33 117 151)(18 34 118 152)(19 35 119 134)(39 64 98 95)(40 65 99 77)(41 66 100 78)(42 67 101 79)(43 68 102 80)(44 69 103 81)(45 70 104 82)(46 71 105 83)(47 72 106 84)(48 73 107 85)(49 74 108 86)(50 75 109 87)(51 76 110 88)(52 58 111 89)(53 59 112 90)(54 60 113 91)(55 61 114 92)(56 62 96 93)(57 63 97 94)
(1 135 120 36)(2 136 121 37)(3 137 122 38)(4 138 123 20)(5 139 124 21)(6 140 125 22)(7 141 126 23)(8 142 127 24)(9 143 128 25)(10 144 129 26)(11 145 130 27)(12 146 131 28)(13 147 132 29)(14 148 133 30)(15 149 115 31)(16 150 116 32)(17 151 117 33)(18 152 118 34)(19 134 119 35)(39 64 98 95)(40 65 99 77)(41 66 100 78)(42 67 101 79)(43 68 102 80)(44 69 103 81)(45 70 104 82)(46 71 105 83)(47 72 106 84)(48 73 107 85)(49 74 108 86)(50 75 109 87)(51 76 110 88)(52 58 111 89)(53 59 112 90)(54 60 113 91)(55 61 114 92)(56 62 96 93)(57 63 97 94)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 67)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(77 136)(78 137)(79 138)(80 139)(81 140)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 134)(95 135)(96 118)(97 119)(98 120)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(111 133)(112 115)(113 116)(114 117)
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), (1,36,120,135)(2,37,121,136)(3,38,122,137)(4,20,123,138)(5,21,124,139)(6,22,125,140)(7,23,126,141)(8,24,127,142)(9,25,128,143)(10,26,129,144)(11,27,130,145)(12,28,131,146)(13,29,132,147)(14,30,133,148)(15,31,115,149)(16,32,116,150)(17,33,117,151)(18,34,118,152)(19,35,119,134)(39,64,98,95)(40,65,99,77)(41,66,100,78)(42,67,101,79)(43,68,102,80)(44,69,103,81)(45,70,104,82)(46,71,105,83)(47,72,106,84)(48,73,107,85)(49,74,108,86)(50,75,109,87)(51,76,110,88)(52,58,111,89)(53,59,112,90)(54,60,113,91)(55,61,114,92)(56,62,96,93)(57,63,97,94), (1,135,120,36)(2,136,121,37)(3,137,122,38)(4,138,123,20)(5,139,124,21)(6,140,125,22)(7,141,126,23)(8,142,127,24)(9,143,128,25)(10,144,129,26)(11,145,130,27)(12,146,131,28)(13,147,132,29)(14,148,133,30)(15,149,115,31)(16,150,116,32)(17,151,117,33)(18,152,118,34)(19,134,119,35)(39,64,98,95)(40,65,99,77)(41,66,100,78)(42,67,101,79)(43,68,102,80)(44,69,103,81)(45,70,104,82)(46,71,105,83)(47,72,106,84)(48,73,107,85)(49,74,108,86)(50,75,109,87)(51,76,110,88)(52,58,111,89)(53,59,112,90)(54,60,113,91)(55,61,114,92)(56,62,96,93)(57,63,97,94), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,134)(95,135)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)(112,115)(113,116)(114,117)>;
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), (1,36,120,135)(2,37,121,136)(3,38,122,137)(4,20,123,138)(5,21,124,139)(6,22,125,140)(7,23,126,141)(8,24,127,142)(9,25,128,143)(10,26,129,144)(11,27,130,145)(12,28,131,146)(13,29,132,147)(14,30,133,148)(15,31,115,149)(16,32,116,150)(17,33,117,151)(18,34,118,152)(19,35,119,134)(39,64,98,95)(40,65,99,77)(41,66,100,78)(42,67,101,79)(43,68,102,80)(44,69,103,81)(45,70,104,82)(46,71,105,83)(47,72,106,84)(48,73,107,85)(49,74,108,86)(50,75,109,87)(51,76,110,88)(52,58,111,89)(53,59,112,90)(54,60,113,91)(55,61,114,92)(56,62,96,93)(57,63,97,94), (1,135,120,36)(2,136,121,37)(3,137,122,38)(4,138,123,20)(5,139,124,21)(6,140,125,22)(7,141,126,23)(8,142,127,24)(9,143,128,25)(10,144,129,26)(11,145,130,27)(12,146,131,28)(13,147,132,29)(14,148,133,30)(15,149,115,31)(16,150,116,32)(17,151,117,33)(18,152,118,34)(19,134,119,35)(39,64,98,95)(40,65,99,77)(41,66,100,78)(42,67,101,79)(43,68,102,80)(44,69,103,81)(45,70,104,82)(46,71,105,83)(47,72,106,84)(48,73,107,85)(49,74,108,86)(50,75,109,87)(51,76,110,88)(52,58,111,89)(53,59,112,90)(54,60,113,91)(55,61,114,92)(56,62,96,93)(57,63,97,94), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,134)(95,135)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)(112,115)(113,116)(114,117) );
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)], [(1,36,120,135),(2,37,121,136),(3,38,122,137),(4,20,123,138),(5,21,124,139),(6,22,125,140),(7,23,126,141),(8,24,127,142),(9,25,128,143),(10,26,129,144),(11,27,130,145),(12,28,131,146),(13,29,132,147),(14,30,133,148),(15,31,115,149),(16,32,116,150),(17,33,117,151),(18,34,118,152),(19,35,119,134),(39,64,98,95),(40,65,99,77),(41,66,100,78),(42,67,101,79),(43,68,102,80),(44,69,103,81),(45,70,104,82),(46,71,105,83),(47,72,106,84),(48,73,107,85),(49,74,108,86),(50,75,109,87),(51,76,110,88),(52,58,111,89),(53,59,112,90),(54,60,113,91),(55,61,114,92),(56,62,96,93),(57,63,97,94)], [(1,135,120,36),(2,136,121,37),(3,137,122,38),(4,138,123,20),(5,139,124,21),(6,140,125,22),(7,141,126,23),(8,142,127,24),(9,143,128,25),(10,144,129,26),(11,145,130,27),(12,146,131,28),(13,147,132,29),(14,148,133,30),(15,149,115,31),(16,150,116,32),(17,151,117,33),(18,152,118,34),(19,134,119,35),(39,64,98,95),(40,65,99,77),(41,66,100,78),(42,67,101,79),(43,68,102,80),(44,69,103,81),(45,70,104,82),(46,71,105,83),(47,72,106,84),(48,73,107,85),(49,74,108,86),(50,75,109,87),(51,76,110,88),(52,58,111,89),(53,59,112,90),(54,60,113,91),(55,61,114,92),(56,62,96,93),(57,63,97,94)], [(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,67),(21,68),(22,69),(23,70),(24,71),(25,72),(26,73),(27,74),(28,75),(29,76),(30,58),(31,59),(32,60),(33,61),(34,62),(35,63),(36,64),(37,65),(38,66),(77,136),(78,137),(79,138),(80,139),(81,140),(82,141),(83,142),(84,143),(85,144),(86,145),(87,146),(88,147),(89,148),(90,149),(91,150),(92,151),(93,152),(94,134),(95,135),(96,118),(97,119),(98,120),(99,121),(100,122),(101,123),(102,124),(103,125),(104,126),(105,127),(106,128),(107,129),(108,130),(109,131),(110,132),(111,133),(112,115),(113,116),(114,117)]])
190 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 19A | ··· | 19R | 38A | ··· | 38R | 38S | ··· | 38BT | 76A | ··· | 76AJ | 76AK | ··· | 76CL |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 76 | ··· | 76 | 76 | ··· | 76 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
190 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C19 | C38 | C38 | C38 | C4oD4 | C4oD4xC19 |
| kernel | C4oD4xC19 | C2xC76 | D4xC19 | Q8xC19 | C4oD4 | C2xC4 | D4 | Q8 | C19 | C1 |
| # reps | 1 | 3 | 3 | 1 | 18 | 54 | 54 | 18 | 2 | 36 |
Matrix representation of C4oD4xC19 ►in GL2(F229) generated by
| 104 | 0 |
| 0 | 104 |
| 122 | 0 |
| 0 | 122 |
| 107 | 0 |
| 146 | 122 |
| 83 | 214 |
| 47 | 146 |
G:=sub<GL(2,GF(229))| [104,0,0,104],[122,0,0,122],[107,146,0,122],[83,47,214,146] >;
C4oD4xC19 in GAP, Magma, Sage, TeX
C_4\circ D_4\times C_{19} % in TeX
G:=Group("C4oD4xC19"); // GroupNames label
G:=SmallGroup(304,40);
// by ID
G=gap.SmallGroup(304,40);
# by ID
G:=PCGroup([5,-2,-2,-2,-19,-2,1541,582]);
// Polycyclic
G:=Group<a,b,c,d|a^19=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations
Export