direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D37, C148⋊2C2, C2.1D74, D74.2C2, Dic37⋊2C2, C74.2C22, C37⋊2(C2×C4), SmallGroup(296,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C4×D37 |
Generators and relations for C4×D37
G = < a,b,c | a4=b37=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D37
►On 148 points
Generators in S148
(1 121 68 77)(2 122 69 78)(3 123 70 79)(4 124 71 80)(5 125 72 81)(6 126 73 82)(7 127 74 83)(8 128 38 84)(9 129 39 85)(10 130 40 86)(11 131 41 87)(12 132 42 88)(13 133 43 89)(14 134 44 90)(15 135 45 91)(16 136 46 92)(17 137 47 93)(18 138 48 94)(19 139 49 95)(20 140 50 96)(21 141 51 97)(22 142 52 98)(23 143 53 99)(24 144 54 100)(25 145 55 101)(26 146 56 102)(27 147 57 103)(28 148 58 104)(29 112 59 105)(30 113 60 106)(31 114 61 107)(32 115 62 108)(33 116 63 109)(34 117 64 110)(35 118 65 111)(36 119 66 75)(37 120 67 76)
(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)
(1 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(38 60)(39 59)(40 58)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(75 78)(76 77)(79 111)(80 110)(81 109)(82 108)(83 107)(84 106)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(112 129)(113 128)(114 127)(115 126)(116 125)(117 124)(118 123)(119 122)(120 121)(130 148)(131 147)(132 146)(133 145)(134 144)(135 143)(136 142)(137 141)(138 140)
G:=sub<Sym(148)| (1,121,68,77)(2,122,69,78)(3,123,70,79)(4,124,71,80)(5,125,72,81)(6,126,73,82)(7,127,74,83)(8,128,38,84)(9,129,39,85)(10,130,40,86)(11,131,41,87)(12,132,42,88)(13,133,43,89)(14,134,44,90)(15,135,45,91)(16,136,46,92)(17,137,47,93)(18,138,48,94)(19,139,49,95)(20,140,50,96)(21,141,51,97)(22,142,52,98)(23,143,53,99)(24,144,54,100)(25,145,55,101)(26,146,56,102)(27,147,57,103)(28,148,58,104)(29,112,59,105)(30,113,60,106)(31,114,61,107)(32,115,62,108)(33,116,63,109)(34,117,64,110)(35,118,65,111)(36,119,66,75)(37,120,67,76), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(38,60)(39,59)(40,58)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(75,78)(76,77)(79,111)(80,110)(81,109)(82,108)(83,107)(84,106)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(112,129)(113,128)(114,127)(115,126)(116,125)(117,124)(118,123)(119,122)(120,121)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,140)>;
G:=Group( (1,121,68,77)(2,122,69,78)(3,123,70,79)(4,124,71,80)(5,125,72,81)(6,126,73,82)(7,127,74,83)(8,128,38,84)(9,129,39,85)(10,130,40,86)(11,131,41,87)(12,132,42,88)(13,133,43,89)(14,134,44,90)(15,135,45,91)(16,136,46,92)(17,137,47,93)(18,138,48,94)(19,139,49,95)(20,140,50,96)(21,141,51,97)(22,142,52,98)(23,143,53,99)(24,144,54,100)(25,145,55,101)(26,146,56,102)(27,147,57,103)(28,148,58,104)(29,112,59,105)(30,113,60,106)(31,114,61,107)(32,115,62,108)(33,116,63,109)(34,117,64,110)(35,118,65,111)(36,119,66,75)(37,120,67,76), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(38,60)(39,59)(40,58)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(75,78)(76,77)(79,111)(80,110)(81,109)(82,108)(83,107)(84,106)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(112,129)(113,128)(114,127)(115,126)(116,125)(117,124)(118,123)(119,122)(120,121)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,140) );
G=PermutationGroup([[(1,121,68,77),(2,122,69,78),(3,123,70,79),(4,124,71,80),(5,125,72,81),(6,126,73,82),(7,127,74,83),(8,128,38,84),(9,129,39,85),(10,130,40,86),(11,131,41,87),(12,132,42,88),(13,133,43,89),(14,134,44,90),(15,135,45,91),(16,136,46,92),(17,137,47,93),(18,138,48,94),(19,139,49,95),(20,140,50,96),(21,141,51,97),(22,142,52,98),(23,143,53,99),(24,144,54,100),(25,145,55,101),(26,146,56,102),(27,147,57,103),(28,148,58,104),(29,112,59,105),(30,113,60,106),(31,114,61,107),(32,115,62,108),(33,116,63,109),(34,117,64,110),(35,118,65,111),(36,119,66,75),(37,120,67,76)], [(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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,20),(38,60),(39,59),(40,58),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68),(75,78),(76,77),(79,111),(80,110),(81,109),(82,108),(83,107),(84,106),(85,105),(86,104),(87,103),(88,102),(89,101),(90,100),(91,99),(92,98),(93,97),(94,96),(112,129),(113,128),(114,127),(115,126),(116,125),(117,124),(118,123),(119,122),(120,121),(130,148),(131,147),(132,146),(133,145),(134,144),(135,143),(136,142),(137,141),(138,140)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 37A | ··· | 37R | 74A | ··· | 74R | 148A | ··· | 148AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 37 | ··· | 37 | 74 | ··· | 74 | 148 | ··· | 148 |
| size | 1 | 1 | 37 | 37 | 1 | 1 | 37 | 37 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D37 | D74 | C4×D37 |
| kernel | C4×D37 | Dic37 | C148 | D74 | D37 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 18 | 18 | 36 |
Matrix representation of C4×D37 ►in GL2(𝔽149) generated by
| 44 | 0 |
| 0 | 44 |
| 0 | 1 |
| 148 | 122 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(149))| [44,0,0,44],[0,148,1,122],[0,1,1,0] >;
C4×D37 in GAP, Magma, Sage, TeX
C_4\times D_{37} % in TeX
G:=Group("C4xD37"); // GroupNames label
G:=SmallGroup(296,5);
// by ID
G=gap.SmallGroup(296,5);
# by ID
G:=PCGroup([4,-2,-2,-2,-37,21,4611]);
// Polycyclic
G:=Group<a,b,c|a^4=b^37=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export