direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C5⋊2C16, C10⋊2C16, C20.6C8, C40.8C4, C8.21D10, C8.4Dic5, C40.21C22, C5⋊4(C2×C16), (C2×C8).9D5, (C2×C40).9C2, (C2×C10).4C8, C4.3(C5⋊2C8), C10.17(C2×C8), C20.59(C2×C4), (C2×C20).22C4, (C2×C4).8Dic5, C4.10(C2×Dic5), C22.2(C5⋊2C8), C2.2(C2×C5⋊2C8), SmallGroup(160,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C2×C5⋊2C16 |
Generators and relations for C2×C5⋊2C16
G = < a,b,c | a2=b5=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×C5⋊2C16
►Regular action on 160 points
Generators in S160
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 97)(13 98)(14 99)(15 100)(16 101)(17 124)(18 125)(19 126)(20 127)(21 128)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 120)(30 121)(31 122)(32 123)(33 76)(34 77)(35 78)(36 79)(37 80)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 154)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 145)(57 146)(58 147)(59 148)(60 149)(61 150)(62 151)(63 152)(64 153)(81 144)(82 129)(83 130)(84 131)(85 132)(86 133)(87 134)(88 135)(89 136)(90 137)(91 138)(92 139)(93 140)(94 141)(95 142)(96 143)
(1 125 88 80 148)(2 149 65 89 126)(3 127 90 66 150)(4 151 67 91 128)(5 113 92 68 152)(6 153 69 93 114)(7 115 94 70 154)(8 155 71 95 116)(9 117 96 72 156)(10 157 73 81 118)(11 119 82 74 158)(12 159 75 83 120)(13 121 84 76 160)(14 145 77 85 122)(15 123 86 78 146)(16 147 79 87 124)(17 101 58 36 134)(18 135 37 59 102)(19 103 60 38 136)(20 137 39 61 104)(21 105 62 40 138)(22 139 41 63 106)(23 107 64 42 140)(24 141 43 49 108)(25 109 50 44 142)(26 143 45 51 110)(27 111 52 46 144)(28 129 47 53 112)(29 97 54 48 130)(30 131 33 55 98)(31 99 56 34 132)(32 133 35 57 100)
(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)| (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,97)(13,98)(14,99)(15,100)(16,101)(17,124)(18,125)(19,126)(20,127)(21,128)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,121)(31,122)(32,123)(33,76)(34,77)(35,78)(36,79)(37,80)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,145)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,153)(81,144)(82,129)(83,130)(84,131)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143), (1,125,88,80,148)(2,149,65,89,126)(3,127,90,66,150)(4,151,67,91,128)(5,113,92,68,152)(6,153,69,93,114)(7,115,94,70,154)(8,155,71,95,116)(9,117,96,72,156)(10,157,73,81,118)(11,119,82,74,158)(12,159,75,83,120)(13,121,84,76,160)(14,145,77,85,122)(15,123,86,78,146)(16,147,79,87,124)(17,101,58,36,134)(18,135,37,59,102)(19,103,60,38,136)(20,137,39,61,104)(21,105,62,40,138)(22,139,41,63,106)(23,107,64,42,140)(24,141,43,49,108)(25,109,50,44,142)(26,143,45,51,110)(27,111,52,46,144)(28,129,47,53,112)(29,97,54,48,130)(30,131,33,55,98)(31,99,56,34,132)(32,133,35,57,100), (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( (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,97)(13,98)(14,99)(15,100)(16,101)(17,124)(18,125)(19,126)(20,127)(21,128)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,121)(31,122)(32,123)(33,76)(34,77)(35,78)(36,79)(37,80)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,145)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,153)(81,144)(82,129)(83,130)(84,131)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143), (1,125,88,80,148)(2,149,65,89,126)(3,127,90,66,150)(4,151,67,91,128)(5,113,92,68,152)(6,153,69,93,114)(7,115,94,70,154)(8,155,71,95,116)(9,117,96,72,156)(10,157,73,81,118)(11,119,82,74,158)(12,159,75,83,120)(13,121,84,76,160)(14,145,77,85,122)(15,123,86,78,146)(16,147,79,87,124)(17,101,58,36,134)(18,135,37,59,102)(19,103,60,38,136)(20,137,39,61,104)(21,105,62,40,138)(22,139,41,63,106)(23,107,64,42,140)(24,141,43,49,108)(25,109,50,44,142)(26,143,45,51,110)(27,111,52,46,144)(28,129,47,53,112)(29,97,54,48,130)(30,131,33,55,98)(31,99,56,34,132)(32,133,35,57,100), (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([[(1,102),(2,103),(3,104),(4,105),(5,106),(6,107),(7,108),(8,109),(9,110),(10,111),(11,112),(12,97),(13,98),(14,99),(15,100),(16,101),(17,124),(18,125),(19,126),(20,127),(21,128),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,120),(30,121),(31,122),(32,123),(33,76),(34,77),(35,78),(36,79),(37,80),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,73),(47,74),(48,75),(49,154),(50,155),(51,156),(52,157),(53,158),(54,159),(55,160),(56,145),(57,146),(58,147),(59,148),(60,149),(61,150),(62,151),(63,152),(64,153),(81,144),(82,129),(83,130),(84,131),(85,132),(86,133),(87,134),(88,135),(89,136),(90,137),(91,138),(92,139),(93,140),(94,141),(95,142),(96,143)], [(1,125,88,80,148),(2,149,65,89,126),(3,127,90,66,150),(4,151,67,91,128),(5,113,92,68,152),(6,153,69,93,114),(7,115,94,70,154),(8,155,71,95,116),(9,117,96,72,156),(10,157,73,81,118),(11,119,82,74,158),(12,159,75,83,120),(13,121,84,76,160),(14,145,77,85,122),(15,123,86,78,146),(16,147,79,87,124),(17,101,58,36,134),(18,135,37,59,102),(19,103,60,38,136),(20,137,39,61,104),(21,105,62,40,138),(22,139,41,63,106),(23,107,64,42,140),(24,141,43,49,108),(25,109,50,44,142),(26,143,45,51,110),(27,111,52,46,144),(28,129,47,53,112),(29,97,54,48,130),(30,131,33,55,98),(31,99,56,34,132),(32,133,35,57,100)], [(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⋊2C16 is a maximal subgroup of
C40.10C8 C20⋊3C16 C40.2Q8 C10.SD32 C40.5D4 C10.Q32 C40.7Q8 D40.5C4 C16×Dic5 C40.88D4 C80⋊17C4 D10⋊1C16 D20.4C8 C40.91D4 C40.92D4 C10.D16 C40.15D4 C20.58D8 C5⋊M6(2) D5×C2×C16 D20.5C8 C40.70C23 C40.30C23
C2×C5⋊2C16 is a maximal quotient of
C20⋊3C16 C80.9C4 C40.91D4
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 16A | ··· | 16P | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D5 | Dic5 | D10 | Dic5 | C5⋊2C8 | C5⋊2C8 | C5⋊2C16 |
| kernel | C2×C5⋊2C16 | C5⋊2C16 | C2×C40 | C40 | C2×C20 | C20 | C2×C10 | C10 | C2×C8 | C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 16 |
Matrix representation of C2×C5⋊2C16 ►in GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 240 | 0 |
| 0 | 0 | 240 |
| 1 | 0 | 0 |
| 0 | 189 | 240 |
| 0 | 1 | 0 |
| 197 | 0 | 0 |
| 0 | 170 | 114 |
| 0 | 191 | 71 |
G:=sub<GL(3,GF(241))| [1,0,0,0,240,0,0,0,240],[1,0,0,0,189,1,0,240,0],[197,0,0,0,170,191,0,114,71] >;
C2×C5⋊2C16 in GAP, Magma, Sage, TeX
C_2\times C_5\rtimes_2C_{16} % in TeX
G:=Group("C2xC5:2C16"); // GroupNames label
G:=SmallGroup(160,18);
// by ID
G=gap.SmallGroup(160,18);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,50,69,4613]);
// 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^-1>;
// generators/relations
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