direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C22⋊C4×C19, C22⋊C76, C23.C38, C38.12D4, (C2×C4)⋊1C38, (C2×C38)⋊1C4, (C2×C76)⋊2C2, C2.1(C2×C76), C2.1(D4×C19), C38.10(C2×C4), (C22×C38).1C2, C22.2(C2×C38), (C2×C38).13C22, SmallGroup(304,20)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×C19
G = < a,b,c,d | a19=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Smallest permutation representation of C22⋊C4×C19
►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)
(1 138)(2 139)(3 140)(4 141)(5 142)(6 143)(7 144)(8 145)(9 146)(10 147)(11 148)(12 149)(13 150)(14 151)(15 152)(16 134)(17 135)(18 136)(19 137)(20 126)(21 127)(22 128)(23 129)(24 130)(25 131)(26 132)(27 133)(28 115)(29 116)(30 117)(31 118)(32 119)(33 120)(34 121)(35 122)(36 123)(37 124)(38 125)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(77 97)(78 98)(79 99)(80 100)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 96)
(1 109)(2 110)(3 111)(4 112)(5 113)(6 114)(7 96)(8 97)(9 98)(10 99)(11 100)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 125)(40 126)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 133)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 121)(55 122)(56 123)(57 124)(77 145)(78 146)(79 147)(80 148)(81 149)(82 150)(83 151)(84 152)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(91 140)(92 141)(93 142)(94 143)(95 144)
(1 116 138 58)(2 117 139 59)(3 118 140 60)(4 119 141 61)(5 120 142 62)(6 121 143 63)(7 122 144 64)(8 123 145 65)(9 124 146 66)(10 125 147 67)(11 126 148 68)(12 127 149 69)(13 128 150 70)(14 129 151 71)(15 130 152 72)(16 131 134 73)(17 132 135 74)(18 133 136 75)(19 115 137 76)(20 100 40 80)(21 101 41 81)(22 102 42 82)(23 103 43 83)(24 104 44 84)(25 105 45 85)(26 106 46 86)(27 107 47 87)(28 108 48 88)(29 109 49 89)(30 110 50 90)(31 111 51 91)(32 112 52 92)(33 113 53 93)(34 114 54 94)(35 96 55 95)(36 97 56 77)(37 98 57 78)(38 99 39 79)
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), (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,145)(9,146)(10,147)(11,148)(12,149)(13,150)(14,151)(15,152)(16,134)(17,135)(18,136)(19,137)(20,126)(21,127)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,115)(29,116)(30,117)(31,118)(32,119)(33,120)(34,121)(35,122)(36,123)(37,124)(38,125)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(77,97)(78,98)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,96), (1,109)(2,110)(3,111)(4,112)(5,113)(6,114)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(77,145)(78,146)(79,147)(80,148)(81,149)(82,150)(83,151)(84,152)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144), (1,116,138,58)(2,117,139,59)(3,118,140,60)(4,119,141,61)(5,120,142,62)(6,121,143,63)(7,122,144,64)(8,123,145,65)(9,124,146,66)(10,125,147,67)(11,126,148,68)(12,127,149,69)(13,128,150,70)(14,129,151,71)(15,130,152,72)(16,131,134,73)(17,132,135,74)(18,133,136,75)(19,115,137,76)(20,100,40,80)(21,101,41,81)(22,102,42,82)(23,103,43,83)(24,104,44,84)(25,105,45,85)(26,106,46,86)(27,107,47,87)(28,108,48,88)(29,109,49,89)(30,110,50,90)(31,111,51,91)(32,112,52,92)(33,113,53,93)(34,114,54,94)(35,96,55,95)(36,97,56,77)(37,98,57,78)(38,99,39,79)>;
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), (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,145)(9,146)(10,147)(11,148)(12,149)(13,150)(14,151)(15,152)(16,134)(17,135)(18,136)(19,137)(20,126)(21,127)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,115)(29,116)(30,117)(31,118)(32,119)(33,120)(34,121)(35,122)(36,123)(37,124)(38,125)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(77,97)(78,98)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,96), (1,109)(2,110)(3,111)(4,112)(5,113)(6,114)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(77,145)(78,146)(79,147)(80,148)(81,149)(82,150)(83,151)(84,152)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144), (1,116,138,58)(2,117,139,59)(3,118,140,60)(4,119,141,61)(5,120,142,62)(6,121,143,63)(7,122,144,64)(8,123,145,65)(9,124,146,66)(10,125,147,67)(11,126,148,68)(12,127,149,69)(13,128,150,70)(14,129,151,71)(15,130,152,72)(16,131,134,73)(17,132,135,74)(18,133,136,75)(19,115,137,76)(20,100,40,80)(21,101,41,81)(22,102,42,82)(23,103,43,83)(24,104,44,84)(25,105,45,85)(26,106,46,86)(27,107,47,87)(28,108,48,88)(29,109,49,89)(30,110,50,90)(31,111,51,91)(32,112,52,92)(33,113,53,93)(34,114,54,94)(35,96,55,95)(36,97,56,77)(37,98,57,78)(38,99,39,79) );
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)], [(1,138),(2,139),(3,140),(4,141),(5,142),(6,143),(7,144),(8,145),(9,146),(10,147),(11,148),(12,149),(13,150),(14,151),(15,152),(16,134),(17,135),(18,136),(19,137),(20,126),(21,127),(22,128),(23,129),(24,130),(25,131),(26,132),(27,133),(28,115),(29,116),(30,117),(31,118),(32,119),(33,120),(34,121),(35,122),(36,123),(37,124),(38,125),(39,67),(40,68),(41,69),(42,70),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(77,97),(78,98),(79,99),(80,100),(81,101),(82,102),(83,103),(84,104),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,96)], [(1,109),(2,110),(3,111),(4,112),(5,113),(6,114),(7,96),(8,97),(9,98),(10,99),(11,100),(12,101),(13,102),(14,103),(15,104),(16,105),(17,106),(18,107),(19,108),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,58),(30,59),(31,60),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,125),(40,126),(41,127),(42,128),(43,129),(44,130),(45,131),(46,132),(47,133),(48,115),(49,116),(50,117),(51,118),(52,119),(53,120),(54,121),(55,122),(56,123),(57,124),(77,145),(78,146),(79,147),(80,148),(81,149),(82,150),(83,151),(84,152),(85,134),(86,135),(87,136),(88,137),(89,138),(90,139),(91,140),(92,141),(93,142),(94,143),(95,144)], [(1,116,138,58),(2,117,139,59),(3,118,140,60),(4,119,141,61),(5,120,142,62),(6,121,143,63),(7,122,144,64),(8,123,145,65),(9,124,146,66),(10,125,147,67),(11,126,148,68),(12,127,149,69),(13,128,150,70),(14,129,151,71),(15,130,152,72),(16,131,134,73),(17,132,135,74),(18,133,136,75),(19,115,137,76),(20,100,40,80),(21,101,41,81),(22,102,42,82),(23,103,43,83),(24,104,44,84),(25,105,45,85),(26,106,46,86),(27,107,47,87),(28,108,48,88),(29,109,49,89),(30,110,50,90),(31,111,51,91),(32,112,52,92),(33,113,53,93),(34,114,54,94),(35,96,55,95),(36,97,56,77),(37,98,57,78),(38,99,39,79)]])
190 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 19A | ··· | 19R | 38A | ··· | 38BB | 38BC | ··· | 38CL | 76A | ··· | 76BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
190 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C19 | C38 | C38 | C76 | D4 | D4×C19 |
| kernel | C22⋊C4×C19 | C2×C76 | C22×C38 | C2×C38 | C22⋊C4 | C2×C4 | C23 | C22 | C38 | C2 |
| # reps | 1 | 2 | 1 | 4 | 18 | 36 | 18 | 72 | 2 | 36 |
Matrix representation of C22⋊C4×C19 ►in GL3(𝔽229) generated by
| 1 | 0 | 0 |
| 0 | 43 | 0 |
| 0 | 0 | 43 |
| 1 | 0 | 0 |
| 0 | 228 | 0 |
| 0 | 183 | 1 |
| 1 | 0 | 0 |
| 0 | 228 | 0 |
| 0 | 0 | 228 |
| 107 | 0 | 0 |
| 0 | 46 | 227 |
| 0 | 28 | 183 |
G:=sub<GL(3,GF(229))| [1,0,0,0,43,0,0,0,43],[1,0,0,0,228,183,0,0,1],[1,0,0,0,228,0,0,0,228],[107,0,0,0,46,28,0,227,183] >;
C22⋊C4×C19 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times C_{19} % in TeX
G:=Group("C2^2:C4xC19"); // GroupNames label
G:=SmallGroup(304,20);
// by ID
G=gap.SmallGroup(304,20);
# by ID
G:=PCGroup([5,-2,-2,-19,-2,-2,760,781]);
// Polycyclic
G:=Group<a,b,c,d|a^19=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
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