direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C5⋊C16, C10⋊C16, C20.3C8, C5⋊2(C2×C16), C4.3(C5⋊C8), (C2×C4).9F5, C5⋊2C8.5C4, (C2×C20).9C4, (C2×C10).1C8, C10.5(C2×C8), C4.18(C2×F5), C20.17(C2×C4), C22.2(C5⋊C8), C5⋊2C8.16C22, C2.1(C2×C5⋊C8), (C2×C5⋊2C8).12C2, SmallGroup(160,72)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C2×C5⋊C16 |
Generators and relations for C2×C5⋊C16
G = < a,b,c | a2=b5=c16=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C2×C5⋊C16
►Regular action on 160 points
Generators in S160
(1 160)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 151)(9 152)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 46)(18 47)(19 48)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(49 140)(50 141)(51 142)(52 143)(53 144)(54 129)(55 130)(56 131)(57 132)(58 133)(59 134)(60 135)(61 136)(62 137)(63 138)(64 139)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 123)(82 124)(83 125)(84 126)(85 127)(86 128)(87 113)(88 114)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 121)(96 122)
(1 83 62 21 99)(2 22 84 100 63)(3 101 23 64 85)(4 49 102 86 24)(5 87 50 25 103)(6 26 88 104 51)(7 105 27 52 89)(8 53 106 90 28)(9 91 54 29 107)(10 30 92 108 55)(11 109 31 56 93)(12 57 110 94 32)(13 95 58 17 111)(14 18 96 112 59)(15 97 19 60 81)(16 61 98 82 20)(33 159 136 74 124)(34 75 160 125 137)(35 126 76 138 145)(36 139 127 146 77)(37 147 140 78 128)(38 79 148 113 141)(39 114 80 142 149)(40 143 115 150 65)(41 151 144 66 116)(42 67 152 117 129)(43 118 68 130 153)(44 131 119 154 69)(45 155 132 70 120)(46 71 156 121 133)(47 122 72 134 157)(48 135 123 158 73)
(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)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
G:=sub<Sym(160)| (1,160)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,151)(9,152)(10,153)(11,154)(12,155)(13,156)(14,157)(15,158)(16,159)(17,46)(18,47)(19,48)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(49,140)(50,141)(51,142)(52,143)(53,144)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,121)(96,122), (1,83,62,21,99)(2,22,84,100,63)(3,101,23,64,85)(4,49,102,86,24)(5,87,50,25,103)(6,26,88,104,51)(7,105,27,52,89)(8,53,106,90,28)(9,91,54,29,107)(10,30,92,108,55)(11,109,31,56,93)(12,57,110,94,32)(13,95,58,17,111)(14,18,96,112,59)(15,97,19,60,81)(16,61,98,82,20)(33,159,136,74,124)(34,75,160,125,137)(35,126,76,138,145)(36,139,127,146,77)(37,147,140,78,128)(38,79,148,113,141)(39,114,80,142,149)(40,143,115,150,65)(41,151,144,66,116)(42,67,152,117,129)(43,118,68,130,153)(44,131,119,154,69)(45,155,132,70,120)(46,71,156,121,133)(47,122,72,134,157)(48,135,123,158,73), (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)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)>;
G:=Group( (1,160)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,151)(9,152)(10,153)(11,154)(12,155)(13,156)(14,157)(15,158)(16,159)(17,46)(18,47)(19,48)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(49,140)(50,141)(51,142)(52,143)(53,144)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,121)(96,122), (1,83,62,21,99)(2,22,84,100,63)(3,101,23,64,85)(4,49,102,86,24)(5,87,50,25,103)(6,26,88,104,51)(7,105,27,52,89)(8,53,106,90,28)(9,91,54,29,107)(10,30,92,108,55)(11,109,31,56,93)(12,57,110,94,32)(13,95,58,17,111)(14,18,96,112,59)(15,97,19,60,81)(16,61,98,82,20)(33,159,136,74,124)(34,75,160,125,137)(35,126,76,138,145)(36,139,127,146,77)(37,147,140,78,128)(38,79,148,113,141)(39,114,80,142,149)(40,143,115,150,65)(41,151,144,66,116)(42,67,152,117,129)(43,118,68,130,153)(44,131,119,154,69)(45,155,132,70,120)(46,71,156,121,133)(47,122,72,134,157)(48,135,123,158,73), (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)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160) );
G=PermutationGroup([[(1,160),(2,145),(3,146),(4,147),(5,148),(6,149),(7,150),(8,151),(9,152),(10,153),(11,154),(12,155),(13,156),(14,157),(15,158),(16,159),(17,46),(18,47),(19,48),(20,33),(21,34),(22,35),(23,36),(24,37),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(49,140),(50,141),(51,142),(52,143),(53,144),(54,129),(55,130),(56,131),(57,132),(58,133),(59,134),(60,135),(61,136),(62,137),(63,138),(64,139),(65,105),(66,106),(67,107),(68,108),(69,109),(70,110),(71,111),(72,112),(73,97),(74,98),(75,99),(76,100),(77,101),(78,102),(79,103),(80,104),(81,123),(82,124),(83,125),(84,126),(85,127),(86,128),(87,113),(88,114),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,121),(96,122)], [(1,83,62,21,99),(2,22,84,100,63),(3,101,23,64,85),(4,49,102,86,24),(5,87,50,25,103),(6,26,88,104,51),(7,105,27,52,89),(8,53,106,90,28),(9,91,54,29,107),(10,30,92,108,55),(11,109,31,56,93),(12,57,110,94,32),(13,95,58,17,111),(14,18,96,112,59),(15,97,19,60,81),(16,61,98,82,20),(33,159,136,74,124),(34,75,160,125,137),(35,126,76,138,145),(36,139,127,146,77),(37,147,140,78,128),(38,79,148,113,141),(39,114,80,142,149),(40,143,115,150,65),(41,151,144,66,116),(42,67,152,117,129),(43,118,68,130,153),(44,131,119,154,69),(45,155,132,70,120),(46,71,156,121,133),(47,122,72,134,157),(48,135,123,158,73)], [(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),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)]])
C2×C5⋊C16 is a maximal subgroup of
C20⋊C16 C42.4F5 Dic5⋊C16 C40.C8 D10⋊C16 C10.M5(2) D20.C8 C10.6M5(2) D4.(C5⋊C8) Dic10.C8 C5⋊C16.C22
C2×C5⋊C16 is a maximal quotient of
C20⋊C16 C5⋊M6(2) C10.6M5(2)
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 8A | ··· | 8H | 10A | 10B | 10C | 16A | ··· | 16P | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 5 | ··· | 5 | 4 | 4 | 4 | 5 | ··· | 5 | 4 | 4 | 4 | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | F5 | C5⋊C8 | C2×F5 | C5⋊C8 | C5⋊C16 |
| kernel | C2×C5⋊C16 | C5⋊C16 | C2×C5⋊2C8 | C5⋊2C8 | C2×C20 | C20 | C2×C10 | C10 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 4 |
Matrix representation of C2×C5⋊C16 ►in GL5(𝔽241)
| 240 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 1 | 240 |
| 130 | 0 | 0 | 0 | 0 |
| 0 | 24 | 4 | 29 | 166 |
| 0 | 53 | 170 | 47 | 190 |
| 0 | 71 | 194 | 51 | 219 |
| 0 | 75 | 223 | 217 | 237 |
G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,240,240,240,240],[130,0,0,0,0,0,24,53,71,75,0,4,170,194,223,0,29,47,51,217,0,166,190,219,237] >;
C2×C5⋊C16 in GAP, Magma, Sage, TeX
C_2\times C_5\rtimes C_{16} % in TeX
G:=Group("C2xC5:C16"); // GroupNames label
G:=SmallGroup(160,72);
// by ID
G=gap.SmallGroup(160,72);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,50,69,2309,1169]);
// Polycyclic
G:=Group<a,b,c|a^2=b^5=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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