direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×M5(2), C80⋊14C22, C23.3C40, C40.82C23, (C2×C80)⋊18C2, (C2×C16)⋊8C10, C16⋊4(C2×C10), (C2×C4).6C40, C4.10(C2×C40), C20.84(C2×C8), (C2×C40).57C4, C8.20(C2×C20), (C2×C20).24C8, (C2×C8).14C20, C40.130(C2×C4), C2.6(C22×C40), (C22×C10).8C8, C22.6(C2×C40), (C22×C40).34C2, (C22×C8).16C10, (C22×C4).17C20, C4.35(C22×C20), C8.15(C22×C10), (C22×C20).66C4, C10.59(C22×C8), (C2×C40).448C22, C20.252(C22×C4), (C2×C4).85(C2×C20), (C2×C10).52(C2×C8), (C2×C8).102(C2×C10), (C2×C20).513(C2×C4), SmallGroup(320,1004)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×M5(2)
G = < a,b,c | a10=b16=c2=1, ab=ba, ac=ca, cbc=b9 >
Subgroups: 98 in 90 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, C10, C10, C10, C16, C2×C8, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C16, M5(2), C22×C8, C40, C40, C2×C20, C2×C20, C22×C10, C2×M5(2), C80, C2×C40, C2×C40, C22×C20, C2×C80, C5×M5(2), C22×C40, C10×M5(2)
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, C23, C10, C2×C8, C22×C4, C20, C2×C10, M5(2), C22×C8, C40, C2×C20, C22×C10, C2×M5(2), C2×C40, C22×C20, C5×M5(2), C22×C40, C10×M5(2)
►On 160 points
Generators in S160
(1 49 82 125 71 157 29 97 34 134)(2 50 83 126 72 158 30 98 35 135)(3 51 84 127 73 159 31 99 36 136)(4 52 85 128 74 160 32 100 37 137)(5 53 86 113 75 145 17 101 38 138)(6 54 87 114 76 146 18 102 39 139)(7 55 88 115 77 147 19 103 40 140)(8 56 89 116 78 148 20 104 41 141)(9 57 90 117 79 149 21 105 42 142)(10 58 91 118 80 150 22 106 43 143)(11 59 92 119 65 151 23 107 44 144)(12 60 93 120 66 152 24 108 45 129)(13 61 94 121 67 153 25 109 46 130)(14 62 95 122 68 154 26 110 47 131)(15 63 96 123 69 155 27 111 48 132)(16 64 81 124 70 156 28 112 33 133)
(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)
(1 149)(2 158)(3 151)(4 160)(5 153)(6 146)(7 155)(8 148)(9 157)(10 150)(11 159)(12 152)(13 145)(14 154)(15 147)(16 156)(17 61)(18 54)(19 63)(20 56)(21 49)(22 58)(23 51)(24 60)(25 53)(26 62)(27 55)(28 64)(29 57)(30 50)(31 59)(32 52)(33 124)(34 117)(35 126)(36 119)(37 128)(38 121)(39 114)(40 123)(41 116)(42 125)(43 118)(44 127)(45 120)(46 113)(47 122)(48 115)(65 136)(66 129)(67 138)(68 131)(69 140)(70 133)(71 142)(72 135)(73 144)(74 137)(75 130)(76 139)(77 132)(78 141)(79 134)(80 143)(81 112)(82 105)(83 98)(84 107)(85 100)(86 109)(87 102)(88 111)(89 104)(90 97)(91 106)(92 99)(93 108)(94 101)(95 110)(96 103)
G:=sub<Sym(160)| (1,49,82,125,71,157,29,97,34,134)(2,50,83,126,72,158,30,98,35,135)(3,51,84,127,73,159,31,99,36,136)(4,52,85,128,74,160,32,100,37,137)(5,53,86,113,75,145,17,101,38,138)(6,54,87,114,76,146,18,102,39,139)(7,55,88,115,77,147,19,103,40,140)(8,56,89,116,78,148,20,104,41,141)(9,57,90,117,79,149,21,105,42,142)(10,58,91,118,80,150,22,106,43,143)(11,59,92,119,65,151,23,107,44,144)(12,60,93,120,66,152,24,108,45,129)(13,61,94,121,67,153,25,109,46,130)(14,62,95,122,68,154,26,110,47,131)(15,63,96,123,69,155,27,111,48,132)(16,64,81,124,70,156,28,112,33,133), (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), (1,149)(2,158)(3,151)(4,160)(5,153)(6,146)(7,155)(8,148)(9,157)(10,150)(11,159)(12,152)(13,145)(14,154)(15,147)(16,156)(17,61)(18,54)(19,63)(20,56)(21,49)(22,58)(23,51)(24,60)(25,53)(26,62)(27,55)(28,64)(29,57)(30,50)(31,59)(32,52)(33,124)(34,117)(35,126)(36,119)(37,128)(38,121)(39,114)(40,123)(41,116)(42,125)(43,118)(44,127)(45,120)(46,113)(47,122)(48,115)(65,136)(66,129)(67,138)(68,131)(69,140)(70,133)(71,142)(72,135)(73,144)(74,137)(75,130)(76,139)(77,132)(78,141)(79,134)(80,143)(81,112)(82,105)(83,98)(84,107)(85,100)(86,109)(87,102)(88,111)(89,104)(90,97)(91,106)(92,99)(93,108)(94,101)(95,110)(96,103)>;
G:=Group( (1,49,82,125,71,157,29,97,34,134)(2,50,83,126,72,158,30,98,35,135)(3,51,84,127,73,159,31,99,36,136)(4,52,85,128,74,160,32,100,37,137)(5,53,86,113,75,145,17,101,38,138)(6,54,87,114,76,146,18,102,39,139)(7,55,88,115,77,147,19,103,40,140)(8,56,89,116,78,148,20,104,41,141)(9,57,90,117,79,149,21,105,42,142)(10,58,91,118,80,150,22,106,43,143)(11,59,92,119,65,151,23,107,44,144)(12,60,93,120,66,152,24,108,45,129)(13,61,94,121,67,153,25,109,46,130)(14,62,95,122,68,154,26,110,47,131)(15,63,96,123,69,155,27,111,48,132)(16,64,81,124,70,156,28,112,33,133), (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), (1,149)(2,158)(3,151)(4,160)(5,153)(6,146)(7,155)(8,148)(9,157)(10,150)(11,159)(12,152)(13,145)(14,154)(15,147)(16,156)(17,61)(18,54)(19,63)(20,56)(21,49)(22,58)(23,51)(24,60)(25,53)(26,62)(27,55)(28,64)(29,57)(30,50)(31,59)(32,52)(33,124)(34,117)(35,126)(36,119)(37,128)(38,121)(39,114)(40,123)(41,116)(42,125)(43,118)(44,127)(45,120)(46,113)(47,122)(48,115)(65,136)(66,129)(67,138)(68,131)(69,140)(70,133)(71,142)(72,135)(73,144)(74,137)(75,130)(76,139)(77,132)(78,141)(79,134)(80,143)(81,112)(82,105)(83,98)(84,107)(85,100)(86,109)(87,102)(88,111)(89,104)(90,97)(91,106)(92,99)(93,108)(94,101)(95,110)(96,103) );
G=PermutationGroup([[(1,49,82,125,71,157,29,97,34,134),(2,50,83,126,72,158,30,98,35,135),(3,51,84,127,73,159,31,99,36,136),(4,52,85,128,74,160,32,100,37,137),(5,53,86,113,75,145,17,101,38,138),(6,54,87,114,76,146,18,102,39,139),(7,55,88,115,77,147,19,103,40,140),(8,56,89,116,78,148,20,104,41,141),(9,57,90,117,79,149,21,105,42,142),(10,58,91,118,80,150,22,106,43,143),(11,59,92,119,65,151,23,107,44,144),(12,60,93,120,66,152,24,108,45,129),(13,61,94,121,67,153,25,109,46,130),(14,62,95,122,68,154,26,110,47,131),(15,63,96,123,69,155,27,111,48,132),(16,64,81,124,70,156,28,112,33,133)], [(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)], [(1,149),(2,158),(3,151),(4,160),(5,153),(6,146),(7,155),(8,148),(9,157),(10,150),(11,159),(12,152),(13,145),(14,154),(15,147),(16,156),(17,61),(18,54),(19,63),(20,56),(21,49),(22,58),(23,51),(24,60),(25,53),(26,62),(27,55),(28,64),(29,57),(30,50),(31,59),(32,52),(33,124),(34,117),(35,126),(36,119),(37,128),(38,121),(39,114),(40,123),(41,116),(42,125),(43,118),(44,127),(45,120),(46,113),(47,122),(48,115),(65,136),(66,129),(67,138),(68,131),(69,140),(70,133),(71,142),(72,135),(73,144),(74,137),(75,130),(76,139),(77,132),(78,141),(79,134),(80,143),(81,112),(82,105),(83,98),(84,107),(85,100),(86,109),(87,102),(88,111),(89,104),(90,97),(91,106),(92,99),(93,108),(94,101),(95,110),(96,103)]])
200 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10L | 10M | ··· | 10T | 16A | ··· | 16P | 20A | ··· | 20P | 20Q | ··· | 20X | 40A | ··· | 40AF | 40AG | ··· | 40AV | 80A | ··· | 80BL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C8 | C8 | C10 | C10 | C10 | C20 | C20 | C40 | C40 | M5(2) | C5×M5(2) |
| kernel | C10×M5(2) | C2×C80 | C5×M5(2) | C22×C40 | C2×C40 | C22×C20 | C2×M5(2) | C2×C20 | C22×C10 | C2×C16 | M5(2) | C22×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 4 | 12 | 4 | 8 | 16 | 4 | 24 | 8 | 48 | 16 | 8 | 32 |
Matrix representation of C10×M5(2) ►in GL3(𝔽241) generated by
| 240 | 0 | 0 |
| 0 | 150 | 0 |
| 0 | 0 | 150 |
| 240 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 8 | 0 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 240 |
G:=sub<GL(3,GF(241))| [240,0,0,0,150,0,0,0,150],[240,0,0,0,0,8,0,1,0],[1,0,0,0,1,0,0,0,240] >;
C10×M5(2) in GAP, Magma, Sage, TeX
C_{10}\times M_5(2) % in TeX
G:=Group("C10xM5(2)"); // GroupNames label
G:=SmallGroup(320,1004);
// by ID
G=gap.SmallGroup(320,1004);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,2269,102,124]);
// Polycyclic
G:=Group<a,b,c|a^10=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations