direct product, abelian, monomial, 3-elementary
Aliases: C3×C48, SmallGroup(144,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C48 |
| C1 — C3×C48 |
| C1 — C3×C48 |
Generators and relations for C3×C48
G = < a,b | a3=b48=1, ab=ba >
Smallest permutation representation of C3×C48
►Regular action on 144 points
Generators in S144
(1 77 97)(2 78 98)(3 79 99)(4 80 100)(5 81 101)(6 82 102)(7 83 103)(8 84 104)(9 85 105)(10 86 106)(11 87 107)(12 88 108)(13 89 109)(14 90 110)(15 91 111)(16 92 112)(17 93 113)(18 94 114)(19 95 115)(20 96 116)(21 49 117)(22 50 118)(23 51 119)(24 52 120)(25 53 121)(26 54 122)(27 55 123)(28 56 124)(29 57 125)(30 58 126)(31 59 127)(32 60 128)(33 61 129)(34 62 130)(35 63 131)(36 64 132)(37 65 133)(38 66 134)(39 67 135)(40 68 136)(41 69 137)(42 70 138)(43 71 139)(44 72 140)(45 73 141)(46 74 142)(47 75 143)(48 76 144)
(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)
G:=sub<Sym(144)| (1,77,97)(2,78,98)(3,79,99)(4,80,100)(5,81,101)(6,82,102)(7,83,103)(8,84,104)(9,85,105)(10,86,106)(11,87,107)(12,88,108)(13,89,109)(14,90,110)(15,91,111)(16,92,112)(17,93,113)(18,94,114)(19,95,115)(20,96,116)(21,49,117)(22,50,118)(23,51,119)(24,52,120)(25,53,121)(26,54,122)(27,55,123)(28,56,124)(29,57,125)(30,58,126)(31,59,127)(32,60,128)(33,61,129)(34,62,130)(35,63,131)(36,64,132)(37,65,133)(38,66,134)(39,67,135)(40,68,136)(41,69,137)(42,70,138)(43,71,139)(44,72,140)(45,73,141)(46,74,142)(47,75,143)(48,76,144), (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)>;
G:=Group( (1,77,97)(2,78,98)(3,79,99)(4,80,100)(5,81,101)(6,82,102)(7,83,103)(8,84,104)(9,85,105)(10,86,106)(11,87,107)(12,88,108)(13,89,109)(14,90,110)(15,91,111)(16,92,112)(17,93,113)(18,94,114)(19,95,115)(20,96,116)(21,49,117)(22,50,118)(23,51,119)(24,52,120)(25,53,121)(26,54,122)(27,55,123)(28,56,124)(29,57,125)(30,58,126)(31,59,127)(32,60,128)(33,61,129)(34,62,130)(35,63,131)(36,64,132)(37,65,133)(38,66,134)(39,67,135)(40,68,136)(41,69,137)(42,70,138)(43,71,139)(44,72,140)(45,73,141)(46,74,142)(47,75,143)(48,76,144), (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) );
G=PermutationGroup([[(1,77,97),(2,78,98),(3,79,99),(4,80,100),(5,81,101),(6,82,102),(7,83,103),(8,84,104),(9,85,105),(10,86,106),(11,87,107),(12,88,108),(13,89,109),(14,90,110),(15,91,111),(16,92,112),(17,93,113),(18,94,114),(19,95,115),(20,96,116),(21,49,117),(22,50,118),(23,51,119),(24,52,120),(25,53,121),(26,54,122),(27,55,123),(28,56,124),(29,57,125),(30,58,126),(31,59,127),(32,60,128),(33,61,129),(34,62,130),(35,63,131),(36,64,132),(37,65,133),(38,66,134),(39,67,135),(40,68,136),(41,69,137),(42,70,138),(43,71,139),(44,72,140),(45,73,141),(46,74,142),(47,75,143),(48,76,144)], [(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)]])
C3×C48 is a maximal subgroup of
C48.S3 C48⋊S3 C32⋊5D16 C6.D24 C32⋊5Q32
144 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 16A | ··· | 16H | 24A | ··· | 24AF | 48A | ··· | 48BL |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C48 |
| kernel | C3×C48 | C3×C24 | C48 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 4 | 16 | 8 | 32 | 64 |
Matrix representation of C3×C48 ►in GL2(𝔽97) generated by
| 61 | 0 |
| 0 | 1 |
| 36 | 0 |
| 0 | 48 |
G:=sub<GL(2,GF(97))| [61,0,0,1],[36,0,0,48] >;
C3×C48 in GAP, Magma, Sage, TeX
C_3\times C_{48} % in TeX
G:=Group("C3xC48"); // GroupNames label
G:=SmallGroup(144,30);
// by ID
G=gap.SmallGroup(144,30);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-2,-2,108,69,88]);
// Polycyclic
G:=Group<a,b|a^3=b^48=1,a*b=b*a>;
// generators/relations
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