direct product, abelian, monomial, 2-elementary
Aliases: C2×C74, SmallGroup(148,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C74 |
| C1 — C2×C74 |
| C1 — C2×C74 |
Generators and relations for C2×C74
G = < a,b | a2=b74=1, ab=ba >
Smallest permutation representation of C2×C74
►Regular action on 148 points
Generators in S148
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 129)(56 130)(57 131)(58 132)(59 133)(60 134)(61 135)(62 136)(63 137)(64 138)(65 139)(66 140)(67 141)(68 142)(69 143)(70 144)(71 145)(72 146)(73 147)(74 148)
(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)
G:=sub<Sym(148)| (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,145)(72,146)(73,147)(74,148), (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)>;
G:=Group( (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,145)(72,146)(73,147)(74,148), (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) );
G=PermutationGroup([[(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,81),(8,82),(9,83),(10,84),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,113),(40,114),(41,115),(42,116),(43,117),(44,118),(45,119),(46,120),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,129),(56,130),(57,131),(58,132),(59,133),(60,134),(61,135),(62,136),(63,137),(64,138),(65,139),(66,140),(67,141),(68,142),(69,143),(70,144),(71,145),(72,146),(73,147),(74,148)], [(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)]])
C2×C74 is a maximal subgroup of
C37⋊D4 C37⋊A4
148 conjugacy classes
| class | 1 | 2A | 2B | 2C | 37A | ··· | 37AJ | 74A | ··· | 74DD |
| order | 1 | 2 | 2 | 2 | 37 | ··· | 37 | 74 | ··· | 74 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
148 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C37 | C74 |
| kernel | C2×C74 | C74 | C22 | C2 |
| # reps | 1 | 3 | 36 | 108 |
Matrix representation of C2×C74 ►in GL2(𝔽149) generated by
| 148 | 0 |
| 0 | 1 |
| 107 | 0 |
| 0 | 76 |
G:=sub<GL(2,GF(149))| [148,0,0,1],[107,0,0,76] >;
C2×C74 in GAP, Magma, Sage, TeX
C_2\times C_{74} % in TeX
G:=Group("C2xC74"); // GroupNames label
G:=SmallGroup(148,5);
// by ID
G=gap.SmallGroup(148,5);
# by ID
G:=PCGroup([3,-2,-2,-37]);
// Polycyclic
G:=Group<a,b|a^2=b^74=1,a*b=b*a>;
// generators/relations
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