direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.10D4, C20.209(C2×D4), (C2×C20).194D4, (Q8×C10).20C4, (C2×Q8).6Dic5, (C22×Q8).3D5, (C22×C20).38C4, (C2×Q8).167D10, C10⋊3(C4.10D4), (C22×C4).5Dic5, C20.82(C22⋊C4), (C2×C20).476C23, (C22×C4).155D10, C4.12(C23.D5), C23.33(C2×Dic5), (Q8×C10).202C22, C4.Dic5.46C22, C22.7(C22×Dic5), (C22×C20).202C22, C22.36(C23.D5), (Q8×C2×C10).3C2, C5⋊5(C2×C4.10D4), C4.93(C2×C5⋊D4), (C2×C20).294(C2×C4), (C2×C4).26(C2×Dic5), C2.15(C2×C23.D5), (C2×C4).199(C5⋊D4), C10.120(C2×C22⋊C4), (C2×C4).130(C22×D5), (C2×C4.Dic5).28C2, (C2×C10).299(C22×C4), (C22×C10).205(C2×C4), (C2×C10).180(C22⋊C4), SmallGroup(320,853)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C20.10D4
G = < a,b,c,d | a2=b20=1, c4=b10, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b9, dcd-1=b5c3 >
Subgroups: 318 in 146 conjugacy classes, 71 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], Q8 [×8], C23, C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×4], C20 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×2], C4.10D4 [×4], C2×M4(2) [×2], C22×Q8, C5⋊2C8 [×4], C2×C20 [×2], C2×C20 [×8], C2×C20 [×4], C5×Q8 [×8], C22×C10, C2×C4.10D4, C2×C5⋊2C8 [×2], C4.Dic5 [×4], C4.Dic5 [×2], C22×C20, C22×C20 [×2], Q8×C10 [×4], Q8×C10 [×4], C20.10D4 [×4], C2×C4.Dic5 [×2], Q8×C2×C10, C2×C20.10D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C4.10D4 [×2], C2×C22⋊C4, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C2×C4.10D4, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C20.10D4 [×2], C2×C23.D5, C2×C20.10D4
►On 160 points
Generators in S160
(1 144)(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)(17 160)(18 141)(19 142)(20 143)(21 57)(22 58)(23 59)(24 60)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(61 140)(62 121)(63 122)(64 123)(65 124)(66 125)(67 126)(68 127)(69 128)(70 129)(71 130)(72 131)(73 132)(74 133)(75 134)(76 135)(77 136)(78 137)(79 138)(80 139)(81 110)(82 111)(83 112)(84 113)(85 114)(86 115)(87 116)(88 117)(89 118)(90 119)(91 120)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)(99 108)(100 109)
(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 23 159 44 11 33 149 54)(2 22 160 43 12 32 150 53)(3 21 141 42 13 31 151 52)(4 40 142 41 14 30 152 51)(5 39 143 60 15 29 153 50)(6 38 144 59 16 28 154 49)(7 37 145 58 17 27 155 48)(8 36 146 57 18 26 156 47)(9 35 147 56 19 25 157 46)(10 34 148 55 20 24 158 45)(61 87 135 101 71 97 125 111)(62 86 136 120 72 96 126 110)(63 85 137 119 73 95 127 109)(64 84 138 118 74 94 128 108)(65 83 139 117 75 93 129 107)(66 82 140 116 76 92 130 106)(67 81 121 115 77 91 131 105)(68 100 122 114 78 90 132 104)(69 99 123 113 79 89 133 103)(70 98 124 112 80 88 134 102)
(1 117 6 102 11 107 16 112)(2 106 7 111 12 116 17 101)(3 115 8 120 13 105 18 110)(4 104 9 109 14 114 19 119)(5 113 10 118 15 103 20 108)(21 136 26 121 31 126 36 131)(22 125 27 130 32 135 37 140)(23 134 28 139 33 124 38 129)(24 123 29 128 34 133 39 138)(25 132 30 137 35 122 40 127)(41 73 46 78 51 63 56 68)(42 62 47 67 52 72 57 77)(43 71 48 76 53 61 58 66)(44 80 49 65 54 70 59 75)(45 69 50 74 55 79 60 64)(81 146 86 151 91 156 96 141)(82 155 87 160 92 145 97 150)(83 144 88 149 93 154 98 159)(84 153 89 158 94 143 99 148)(85 142 90 147 95 152 100 157)
G:=sub<Sym(160)| (1,144)(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)(17,160)(18,141)(19,142)(20,143)(21,57)(22,58)(23,59)(24,60)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(61,140)(62,121)(63,122)(64,123)(65,124)(66,125)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,110)(82,111)(83,112)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119)(91,120)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107)(99,108)(100,109), (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), (1,23,159,44,11,33,149,54)(2,22,160,43,12,32,150,53)(3,21,141,42,13,31,151,52)(4,40,142,41,14,30,152,51)(5,39,143,60,15,29,153,50)(6,38,144,59,16,28,154,49)(7,37,145,58,17,27,155,48)(8,36,146,57,18,26,156,47)(9,35,147,56,19,25,157,46)(10,34,148,55,20,24,158,45)(61,87,135,101,71,97,125,111)(62,86,136,120,72,96,126,110)(63,85,137,119,73,95,127,109)(64,84,138,118,74,94,128,108)(65,83,139,117,75,93,129,107)(66,82,140,116,76,92,130,106)(67,81,121,115,77,91,131,105)(68,100,122,114,78,90,132,104)(69,99,123,113,79,89,133,103)(70,98,124,112,80,88,134,102), (1,117,6,102,11,107,16,112)(2,106,7,111,12,116,17,101)(3,115,8,120,13,105,18,110)(4,104,9,109,14,114,19,119)(5,113,10,118,15,103,20,108)(21,136,26,121,31,126,36,131)(22,125,27,130,32,135,37,140)(23,134,28,139,33,124,38,129)(24,123,29,128,34,133,39,138)(25,132,30,137,35,122,40,127)(41,73,46,78,51,63,56,68)(42,62,47,67,52,72,57,77)(43,71,48,76,53,61,58,66)(44,80,49,65,54,70,59,75)(45,69,50,74,55,79,60,64)(81,146,86,151,91,156,96,141)(82,155,87,160,92,145,97,150)(83,144,88,149,93,154,98,159)(84,153,89,158,94,143,99,148)(85,142,90,147,95,152,100,157)>;
G:=Group( (1,144)(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)(17,160)(18,141)(19,142)(20,143)(21,57)(22,58)(23,59)(24,60)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(61,140)(62,121)(63,122)(64,123)(65,124)(66,125)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,110)(82,111)(83,112)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119)(91,120)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107)(99,108)(100,109), (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), (1,23,159,44,11,33,149,54)(2,22,160,43,12,32,150,53)(3,21,141,42,13,31,151,52)(4,40,142,41,14,30,152,51)(5,39,143,60,15,29,153,50)(6,38,144,59,16,28,154,49)(7,37,145,58,17,27,155,48)(8,36,146,57,18,26,156,47)(9,35,147,56,19,25,157,46)(10,34,148,55,20,24,158,45)(61,87,135,101,71,97,125,111)(62,86,136,120,72,96,126,110)(63,85,137,119,73,95,127,109)(64,84,138,118,74,94,128,108)(65,83,139,117,75,93,129,107)(66,82,140,116,76,92,130,106)(67,81,121,115,77,91,131,105)(68,100,122,114,78,90,132,104)(69,99,123,113,79,89,133,103)(70,98,124,112,80,88,134,102), (1,117,6,102,11,107,16,112)(2,106,7,111,12,116,17,101)(3,115,8,120,13,105,18,110)(4,104,9,109,14,114,19,119)(5,113,10,118,15,103,20,108)(21,136,26,121,31,126,36,131)(22,125,27,130,32,135,37,140)(23,134,28,139,33,124,38,129)(24,123,29,128,34,133,39,138)(25,132,30,137,35,122,40,127)(41,73,46,78,51,63,56,68)(42,62,47,67,52,72,57,77)(43,71,48,76,53,61,58,66)(44,80,49,65,54,70,59,75)(45,69,50,74,55,79,60,64)(81,146,86,151,91,156,96,141)(82,155,87,160,92,145,97,150)(83,144,88,149,93,154,98,159)(84,153,89,158,94,143,99,148)(85,142,90,147,95,152,100,157) );
G=PermutationGroup([(1,144),(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),(17,160),(18,141),(19,142),(20,143),(21,57),(22,58),(23,59),(24,60),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(61,140),(62,121),(63,122),(64,123),(65,124),(66,125),(67,126),(68,127),(69,128),(70,129),(71,130),(72,131),(73,132),(74,133),(75,134),(76,135),(77,136),(78,137),(79,138),(80,139),(81,110),(82,111),(83,112),(84,113),(85,114),(86,115),(87,116),(88,117),(89,118),(90,119),(91,120),(92,101),(93,102),(94,103),(95,104),(96,105),(97,106),(98,107),(99,108),(100,109)], [(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)], [(1,23,159,44,11,33,149,54),(2,22,160,43,12,32,150,53),(3,21,141,42,13,31,151,52),(4,40,142,41,14,30,152,51),(5,39,143,60,15,29,153,50),(6,38,144,59,16,28,154,49),(7,37,145,58,17,27,155,48),(8,36,146,57,18,26,156,47),(9,35,147,56,19,25,157,46),(10,34,148,55,20,24,158,45),(61,87,135,101,71,97,125,111),(62,86,136,120,72,96,126,110),(63,85,137,119,73,95,127,109),(64,84,138,118,74,94,128,108),(65,83,139,117,75,93,129,107),(66,82,140,116,76,92,130,106),(67,81,121,115,77,91,131,105),(68,100,122,114,78,90,132,104),(69,99,123,113,79,89,133,103),(70,98,124,112,80,88,134,102)], [(1,117,6,102,11,107,16,112),(2,106,7,111,12,116,17,101),(3,115,8,120,13,105,18,110),(4,104,9,109,14,114,19,119),(5,113,10,118,15,103,20,108),(21,136,26,121,31,126,36,131),(22,125,27,130,32,135,37,140),(23,134,28,139,33,124,38,129),(24,123,29,128,34,133,39,138),(25,132,30,137,35,122,40,127),(41,73,46,78,51,63,56,68),(42,62,47,67,52,72,57,77),(43,71,48,76,53,61,58,66),(44,80,49,65,54,70,59,75),(45,69,50,74,55,79,60,64),(81,146,86,151,91,156,96,141),(82,155,87,160,92,145,97,150),(83,144,88,149,93,154,98,159),(84,153,89,158,94,143,99,148),(85,142,90,147,95,152,100,157)])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 20 | ··· | 20 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D5 | Dic5 | D10 | Dic5 | D10 | C5⋊D4 | C4.10D4 | C20.10D4 |
| kernel | C2×C20.10D4 | C20.10D4 | C2×C4.Dic5 | Q8×C2×C10 | C22×C20 | Q8×C10 | C2×C20 | C22×Q8 | C22×C4 | C22×C4 | C2×Q8 | C2×Q8 | C2×C4 | C10 | C2 |
| # reps | 1 | 4 | 2 | 1 | 4 | 4 | 4 | 2 | 4 | 2 | 4 | 4 | 16 | 2 | 8 |
Matrix representation of C2×C20.10D4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 6 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 19 | 5 | 0 | 31 |
| 0 | 0 | 0 | 14 | 10 | 0 |
| 5 | 33 | 0 | 0 | 0 | 0 |
| 3 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 34 | 36 | 0 |
| 0 | 0 | 0 | 2 | 0 | 36 |
| 0 | 0 | 8 | 9 | 32 | 7 |
| 0 | 0 | 33 | 9 | 0 | 39 |
| 36 | 8 | 0 | 0 | 0 | 0 |
| 38 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 30 | 23 | 26 |
| 0 | 0 | 27 | 21 | 26 | 18 |
| 0 | 0 | 15 | 4 | 5 | 38 |
| 0 | 0 | 23 | 9 | 35 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[6,40,0,0,0,0,1,0,0,0,0,0,0,0,0,4,19,0,0,0,37,0,5,14,0,0,0,0,0,10,0,0,0,0,31,0],[5,3,0,0,0,0,33,36,0,0,0,0,0,0,9,0,8,33,0,0,34,2,9,9,0,0,36,0,32,0,0,0,0,36,7,39],[36,38,0,0,0,0,8,5,0,0,0,0,0,0,16,27,15,23,0,0,30,21,4,9,0,0,23,26,5,35,0,0,26,18,38,40] >;
C2×C20.10D4 in GAP, Magma, Sage, TeX
C_2\times C_{20}._{10}D_4 % in TeX
G:=Group("C2xC20.10D4"); // GroupNames label
G:=SmallGroup(320,853);
// by ID
G=gap.SmallGroup(320,853);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,184,297,136,1684,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=1,c^4=b^10,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^9,d*c*d^-1=b^5*c^3>;
// generators/relations