metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C16⋊2D9, C144⋊2C2, C9⋊1SD32, C48.4S3, C2.4D72, C18.2D8, C4.2D36, C6.2D24, D72.1C2, C36.25D4, C8.14D18, C24.68D6, Dic36⋊1C2, C12.35D12, C72.15C22, C3.(C48⋊C2), SmallGroup(288,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C144⋊C2
G = < a,b | a144=b2=1, bab=a71 >
Smallest permutation representation of C144⋊C2
►On 144 points
Generators in S144
(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)
(2 72)(3 143)(4 70)(5 141)(6 68)(7 139)(8 66)(9 137)(10 64)(11 135)(12 62)(13 133)(14 60)(15 131)(16 58)(17 129)(18 56)(19 127)(20 54)(21 125)(22 52)(23 123)(24 50)(25 121)(26 48)(27 119)(28 46)(29 117)(30 44)(31 115)(32 42)(33 113)(34 40)(35 111)(36 38)(37 109)(39 107)(41 105)(43 103)(45 101)(47 99)(49 97)(51 95)(53 93)(55 91)(57 89)(59 87)(61 85)(63 83)(65 81)(67 79)(69 77)(71 75)(74 144)(76 142)(78 140)(80 138)(82 136)(84 134)(86 132)(88 130)(90 128)(92 126)(94 124)(96 122)(98 120)(100 118)(102 116)(104 114)(106 112)(108 110)
G:=sub<Sym(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), (2,72)(3,143)(4,70)(5,141)(6,68)(7,139)(8,66)(9,137)(10,64)(11,135)(12,62)(13,133)(14,60)(15,131)(16,58)(17,129)(18,56)(19,127)(20,54)(21,125)(22,52)(23,123)(24,50)(25,121)(26,48)(27,119)(28,46)(29,117)(30,44)(31,115)(32,42)(33,113)(34,40)(35,111)(36,38)(37,109)(39,107)(41,105)(43,103)(45,101)(47,99)(49,97)(51,95)(53,93)(55,91)(57,89)(59,87)(61,85)(63,83)(65,81)(67,79)(69,77)(71,75)(74,144)(76,142)(78,140)(80,138)(82,136)(84,134)(86,132)(88,130)(90,128)(92,126)(94,124)(96,122)(98,120)(100,118)(102,116)(104,114)(106,112)(108,110)>;
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), (2,72)(3,143)(4,70)(5,141)(6,68)(7,139)(8,66)(9,137)(10,64)(11,135)(12,62)(13,133)(14,60)(15,131)(16,58)(17,129)(18,56)(19,127)(20,54)(21,125)(22,52)(23,123)(24,50)(25,121)(26,48)(27,119)(28,46)(29,117)(30,44)(31,115)(32,42)(33,113)(34,40)(35,111)(36,38)(37,109)(39,107)(41,105)(43,103)(45,101)(47,99)(49,97)(51,95)(53,93)(55,91)(57,89)(59,87)(61,85)(63,83)(65,81)(67,79)(69,77)(71,75)(74,144)(76,142)(78,140)(80,138)(82,136)(84,134)(86,132)(88,130)(90,128)(92,126)(94,124)(96,122)(98,120)(100,118)(102,116)(104,114)(106,112)(108,110) );
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)], [(2,72),(3,143),(4,70),(5,141),(6,68),(7,139),(8,66),(9,137),(10,64),(11,135),(12,62),(13,133),(14,60),(15,131),(16,58),(17,129),(18,56),(19,127),(20,54),(21,125),(22,52),(23,123),(24,50),(25,121),(26,48),(27,119),(28,46),(29,117),(30,44),(31,115),(32,42),(33,113),(34,40),(35,111),(36,38),(37,109),(39,107),(41,105),(43,103),(45,101),(47,99),(49,97),(51,95),(53,93),(55,91),(57,89),(59,87),(61,85),(63,83),(65,81),(67,79),(69,77),(71,75),(74,144),(76,142),(78,140),(80,138),(82,136),(84,134),(86,132),(88,130),(90,128),(92,126),(94,124),(96,122),(98,120),(100,118),(102,116),(104,114),(106,112),(108,110)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6 | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 16A | 16B | 16C | 16D | 18A | 18B | 18C | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 48A | ··· | 48H | 72A | ··· | 72L | 144A | ··· | 144X |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 16 | 16 | 16 | 16 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 48 | ··· | 48 | 72 | ··· | 72 | 144 | ··· | 144 |
| size | 1 | 1 | 72 | 2 | 2 | 72 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D8 | D9 | D12 | SD32 | D18 | D24 | D36 | C48⋊C2 | D72 | C144⋊C2 |
| kernel | C144⋊C2 | C144 | Dic36 | D72 | C48 | C36 | C24 | C18 | C16 | C12 | C9 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 4 | 3 | 4 | 6 | 8 | 12 | 24 |
Matrix representation of C144⋊C2 ►in GL2(𝔽433) generated by
| 287 | 244 |
| 189 | 43 |
| 1 | 0 |
| 432 | 432 |
G:=sub<GL(2,GF(433))| [287,189,244,43],[1,432,0,432] >;
C144⋊C2 in GAP, Magma, Sage, TeX
C_{144}\rtimes C_2 % in TeX
G:=Group("C144:C2"); // GroupNames label
G:=SmallGroup(288,7);
// by ID
G=gap.SmallGroup(288,7);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,590,58,675,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b|a^144=b^2=1,b*a*b=a^71>;
// generators/relations
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