direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C4×C37⋊C3, C148⋊C3, C37⋊4C12, C74.2C6, C2.(C2×C37⋊C3), (C2×C37⋊C3).2C2, SmallGroup(444,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C4×C37⋊C3 |
Generators and relations for C4×C37⋊C3
G = < a,b,c | a4=b37=c3=1, ab=ba, ac=ca, cbc-1=b10 >
Smallest permutation representation of C4×C37⋊C3
►On 148 points
Generators in S148
(1 112 38 75)(2 113 39 76)(3 114 40 77)(4 115 41 78)(5 116 42 79)(6 117 43 80)(7 118 44 81)(8 119 45 82)(9 120 46 83)(10 121 47 84)(11 122 48 85)(12 123 49 86)(13 124 50 87)(14 125 51 88)(15 126 52 89)(16 127 53 90)(17 128 54 91)(18 129 55 92)(19 130 56 93)(20 131 57 94)(21 132 58 95)(22 133 59 96)(23 134 60 97)(24 135 61 98)(25 136 62 99)(26 137 63 100)(27 138 64 101)(28 139 65 102)(29 140 66 103)(30 141 67 104)(31 142 68 105)(32 143 69 106)(33 144 70 107)(34 145 71 108)(35 146 72 109)(36 147 73 110)(37 148 74 111)
(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)
(2 27 11)(3 16 21)(4 5 31)(6 20 14)(7 9 24)(8 35 34)(10 13 17)(12 28 37)(15 32 30)(18 36 23)(19 25 33)(22 29 26)(39 64 48)(40 53 58)(41 42 68)(43 57 51)(44 46 61)(45 72 71)(47 50 54)(49 65 74)(52 69 67)(55 73 60)(56 62 70)(59 66 63)(76 101 85)(77 90 95)(78 79 105)(80 94 88)(81 83 98)(82 109 108)(84 87 91)(86 102 111)(89 106 104)(92 110 97)(93 99 107)(96 103 100)(113 138 122)(114 127 132)(115 116 142)(117 131 125)(118 120 135)(119 146 145)(121 124 128)(123 139 148)(126 143 141)(129 147 134)(130 136 144)(133 140 137)
G:=sub<Sym(148)| (1,112,38,75)(2,113,39,76)(3,114,40,77)(4,115,41,78)(5,116,42,79)(6,117,43,80)(7,118,44,81)(8,119,45,82)(9,120,46,83)(10,121,47,84)(11,122,48,85)(12,123,49,86)(13,124,50,87)(14,125,51,88)(15,126,52,89)(16,127,53,90)(17,128,54,91)(18,129,55,92)(19,130,56,93)(20,131,57,94)(21,132,58,95)(22,133,59,96)(23,134,60,97)(24,135,61,98)(25,136,62,99)(26,137,63,100)(27,138,64,101)(28,139,65,102)(29,140,66,103)(30,141,67,104)(31,142,68,105)(32,143,69,106)(33,144,70,107)(34,145,71,108)(35,146,72,109)(36,147,73,110)(37,148,74,111), (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), (2,27,11)(3,16,21)(4,5,31)(6,20,14)(7,9,24)(8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)(39,64,48)(40,53,58)(41,42,68)(43,57,51)(44,46,61)(45,72,71)(47,50,54)(49,65,74)(52,69,67)(55,73,60)(56,62,70)(59,66,63)(76,101,85)(77,90,95)(78,79,105)(80,94,88)(81,83,98)(82,109,108)(84,87,91)(86,102,111)(89,106,104)(92,110,97)(93,99,107)(96,103,100)(113,138,122)(114,127,132)(115,116,142)(117,131,125)(118,120,135)(119,146,145)(121,124,128)(123,139,148)(126,143,141)(129,147,134)(130,136,144)(133,140,137)>;
G:=Group( (1,112,38,75)(2,113,39,76)(3,114,40,77)(4,115,41,78)(5,116,42,79)(6,117,43,80)(7,118,44,81)(8,119,45,82)(9,120,46,83)(10,121,47,84)(11,122,48,85)(12,123,49,86)(13,124,50,87)(14,125,51,88)(15,126,52,89)(16,127,53,90)(17,128,54,91)(18,129,55,92)(19,130,56,93)(20,131,57,94)(21,132,58,95)(22,133,59,96)(23,134,60,97)(24,135,61,98)(25,136,62,99)(26,137,63,100)(27,138,64,101)(28,139,65,102)(29,140,66,103)(30,141,67,104)(31,142,68,105)(32,143,69,106)(33,144,70,107)(34,145,71,108)(35,146,72,109)(36,147,73,110)(37,148,74,111), (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), (2,27,11)(3,16,21)(4,5,31)(6,20,14)(7,9,24)(8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)(39,64,48)(40,53,58)(41,42,68)(43,57,51)(44,46,61)(45,72,71)(47,50,54)(49,65,74)(52,69,67)(55,73,60)(56,62,70)(59,66,63)(76,101,85)(77,90,95)(78,79,105)(80,94,88)(81,83,98)(82,109,108)(84,87,91)(86,102,111)(89,106,104)(92,110,97)(93,99,107)(96,103,100)(113,138,122)(114,127,132)(115,116,142)(117,131,125)(118,120,135)(119,146,145)(121,124,128)(123,139,148)(126,143,141)(129,147,134)(130,136,144)(133,140,137) );
G=PermutationGroup([[(1,112,38,75),(2,113,39,76),(3,114,40,77),(4,115,41,78),(5,116,42,79),(6,117,43,80),(7,118,44,81),(8,119,45,82),(9,120,46,83),(10,121,47,84),(11,122,48,85),(12,123,49,86),(13,124,50,87),(14,125,51,88),(15,126,52,89),(16,127,53,90),(17,128,54,91),(18,129,55,92),(19,130,56,93),(20,131,57,94),(21,132,58,95),(22,133,59,96),(23,134,60,97),(24,135,61,98),(25,136,62,99),(26,137,63,100),(27,138,64,101),(28,139,65,102),(29,140,66,103),(30,141,67,104),(31,142,68,105),(32,143,69,106),(33,144,70,107),(34,145,71,108),(35,146,72,109),(36,147,73,110),(37,148,74,111)], [(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)], [(2,27,11),(3,16,21),(4,5,31),(6,20,14),(7,9,24),(8,35,34),(10,13,17),(12,28,37),(15,32,30),(18,36,23),(19,25,33),(22,29,26),(39,64,48),(40,53,58),(41,42,68),(43,57,51),(44,46,61),(45,72,71),(47,50,54),(49,65,74),(52,69,67),(55,73,60),(56,62,70),(59,66,63),(76,101,85),(77,90,95),(78,79,105),(80,94,88),(81,83,98),(82,109,108),(84,87,91),(86,102,111),(89,106,104),(92,110,97),(93,99,107),(96,103,100),(113,138,122),(114,127,132),(115,116,142),(117,131,125),(118,120,135),(119,146,145),(121,124,128),(123,139,148),(126,143,141),(129,147,134),(130,136,144),(133,140,137)]])
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 37A | ··· | 37L | 74A | ··· | 74L | 148A | ··· | 148X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 37 | ··· | 37 | 74 | ··· | 74 | 148 | ··· | 148 |
| size | 1 | 1 | 37 | 37 | 1 | 1 | 37 | 37 | 37 | 37 | 37 | 37 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | + | |||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | C37⋊C3 | C2×C37⋊C3 | C4×C37⋊C3 |
| kernel | C4×C37⋊C3 | C2×C37⋊C3 | C148 | C37⋊C3 | C74 | C37 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 12 | 12 | 24 |
Matrix representation of C4×C37⋊C3 ►in GL4(𝔽1777) generated by
| 775 | 0 | 0 | 0 |
| 0 | 1776 | 0 | 0 |
| 0 | 0 | 1776 | 0 |
| 0 | 0 | 0 | 1776 |
| 1 | 0 | 0 | 0 |
| 0 | 1481 | 301 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 629 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 539 | 843 | 1189 |
| 0 | 352 | 1207 | 933 |
G:=sub<GL(4,GF(1777))| [775,0,0,0,0,1776,0,0,0,0,1776,0,0,0,0,1776],[1,0,0,0,0,1481,1,0,0,301,0,1,0,1,0,0],[629,0,0,0,0,1,539,352,0,0,843,1207,0,0,1189,933] >;
C4×C37⋊C3 in GAP, Magma, Sage, TeX
C_4\times C_{37}\rtimes C_3 % in TeX
G:=Group("C4xC37:C3"); // GroupNames label
G:=SmallGroup(444,2);
// by ID
G=gap.SmallGroup(444,2);
# by ID
G:=PCGroup([4,-2,-3,-2,-37,24,2503]);
// Polycyclic
G:=Group<a,b,c|a^4=b^37=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations
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