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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C20, C20, C2×C10, C2×C10, C4.10D4, C2×M4(2), C22×Q8, C5⋊2C8, C2×C20, C2×C20, C2×C20, C5×Q8, C22×C10, C2×C4.10D4, C2×C5⋊2C8, C4.Dic5, C4.Dic5, C22×C20, C22×C20, Q8×C10, Q8×C10, C20.10D4, C2×C4.Dic5, Q8×C2×C10, C2×C20.10D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, Dic5, D10, C4.10D4, C2×C22⋊C4, C2×Dic5, C5⋊D4, C22×D5, C2×C4.10D4, C23.D5, C22×Dic5, C2×C5⋊D4, C20.10D4, C2×C23.D5, C2×C20.10D4
►On 160 points
Generators in S160
(1 146)(2 147)(3 148)(4 149)(5 150)(6 151)(7 152)(8 153)(9 154)(10 155)(11 156)(12 157)(13 158)(14 159)(15 160)(16 141)(17 142)(18 143)(19 144)(20 145)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 41)(40 42)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 81)(101 138)(102 139)(103 140)(104 121)(105 122)(106 123)(107 124)(108 125)(109 126)(110 127)(111 128)(112 129)(113 130)(114 131)(115 132)(116 133)(117 134)(118 135)(119 136)(120 137)
(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 64 141 90 11 74 151 100)(2 63 142 89 12 73 152 99)(3 62 143 88 13 72 153 98)(4 61 144 87 14 71 154 97)(5 80 145 86 15 70 155 96)(6 79 146 85 16 69 156 95)(7 78 147 84 17 68 157 94)(8 77 148 83 18 67 158 93)(9 76 149 82 19 66 159 92)(10 75 150 81 20 65 160 91)(21 113 48 125 31 103 58 135)(22 112 49 124 32 102 59 134)(23 111 50 123 33 101 60 133)(24 110 51 122 34 120 41 132)(25 109 52 121 35 119 42 131)(26 108 53 140 36 118 43 130)(27 107 54 139 37 117 44 129)(28 106 55 138 38 116 45 128)(29 105 56 137 39 115 46 127)(30 104 57 136 40 114 47 126)
(1 33 6 38 11 23 16 28)(2 22 7 27 12 32 17 37)(3 31 8 36 13 21 18 26)(4 40 9 25 14 30 19 35)(5 29 10 34 15 39 20 24)(41 145 46 150 51 155 56 160)(42 154 47 159 52 144 57 149)(43 143 48 148 53 153 58 158)(44 152 49 157 54 142 59 147)(45 141 50 146 55 151 60 156)(61 131 66 136 71 121 76 126)(62 140 67 125 72 130 77 135)(63 129 68 134 73 139 78 124)(64 138 69 123 74 128 79 133)(65 127 70 132 75 137 80 122)(81 105 86 110 91 115 96 120)(82 114 87 119 92 104 97 109)(83 103 88 108 93 113 98 118)(84 112 89 117 94 102 99 107)(85 101 90 106 95 111 100 116)
G:=sub<Sym(160)| (1,146)(2,147)(3,148)(4,149)(5,150)(6,151)(7,152)(8,153)(9,154)(10,155)(11,156)(12,157)(13,158)(14,159)(15,160)(16,141)(17,142)(18,143)(19,144)(20,145)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,41)(40,42)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,81)(101,138)(102,139)(103,140)(104,121)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)(112,129)(113,130)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136)(120,137), (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), (1,64,141,90,11,74,151,100)(2,63,142,89,12,73,152,99)(3,62,143,88,13,72,153,98)(4,61,144,87,14,71,154,97)(5,80,145,86,15,70,155,96)(6,79,146,85,16,69,156,95)(7,78,147,84,17,68,157,94)(8,77,148,83,18,67,158,93)(9,76,149,82,19,66,159,92)(10,75,150,81,20,65,160,91)(21,113,48,125,31,103,58,135)(22,112,49,124,32,102,59,134)(23,111,50,123,33,101,60,133)(24,110,51,122,34,120,41,132)(25,109,52,121,35,119,42,131)(26,108,53,140,36,118,43,130)(27,107,54,139,37,117,44,129)(28,106,55,138,38,116,45,128)(29,105,56,137,39,115,46,127)(30,104,57,136,40,114,47,126), (1,33,6,38,11,23,16,28)(2,22,7,27,12,32,17,37)(3,31,8,36,13,21,18,26)(4,40,9,25,14,30,19,35)(5,29,10,34,15,39,20,24)(41,145,46,150,51,155,56,160)(42,154,47,159,52,144,57,149)(43,143,48,148,53,153,58,158)(44,152,49,157,54,142,59,147)(45,141,50,146,55,151,60,156)(61,131,66,136,71,121,76,126)(62,140,67,125,72,130,77,135)(63,129,68,134,73,139,78,124)(64,138,69,123,74,128,79,133)(65,127,70,132,75,137,80,122)(81,105,86,110,91,115,96,120)(82,114,87,119,92,104,97,109)(83,103,88,108,93,113,98,118)(84,112,89,117,94,102,99,107)(85,101,90,106,95,111,100,116)>;
G:=Group( (1,146)(2,147)(3,148)(4,149)(5,150)(6,151)(7,152)(8,153)(9,154)(10,155)(11,156)(12,157)(13,158)(14,159)(15,160)(16,141)(17,142)(18,143)(19,144)(20,145)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,41)(40,42)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,81)(101,138)(102,139)(103,140)(104,121)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)(112,129)(113,130)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136)(120,137), (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), (1,64,141,90,11,74,151,100)(2,63,142,89,12,73,152,99)(3,62,143,88,13,72,153,98)(4,61,144,87,14,71,154,97)(5,80,145,86,15,70,155,96)(6,79,146,85,16,69,156,95)(7,78,147,84,17,68,157,94)(8,77,148,83,18,67,158,93)(9,76,149,82,19,66,159,92)(10,75,150,81,20,65,160,91)(21,113,48,125,31,103,58,135)(22,112,49,124,32,102,59,134)(23,111,50,123,33,101,60,133)(24,110,51,122,34,120,41,132)(25,109,52,121,35,119,42,131)(26,108,53,140,36,118,43,130)(27,107,54,139,37,117,44,129)(28,106,55,138,38,116,45,128)(29,105,56,137,39,115,46,127)(30,104,57,136,40,114,47,126), (1,33,6,38,11,23,16,28)(2,22,7,27,12,32,17,37)(3,31,8,36,13,21,18,26)(4,40,9,25,14,30,19,35)(5,29,10,34,15,39,20,24)(41,145,46,150,51,155,56,160)(42,154,47,159,52,144,57,149)(43,143,48,148,53,153,58,158)(44,152,49,157,54,142,59,147)(45,141,50,146,55,151,60,156)(61,131,66,136,71,121,76,126)(62,140,67,125,72,130,77,135)(63,129,68,134,73,139,78,124)(64,138,69,123,74,128,79,133)(65,127,70,132,75,137,80,122)(81,105,86,110,91,115,96,120)(82,114,87,119,92,104,97,109)(83,103,88,108,93,113,98,118)(84,112,89,117,94,102,99,107)(85,101,90,106,95,111,100,116) );
G=PermutationGroup([[(1,146),(2,147),(3,148),(4,149),(5,150),(6,151),(7,152),(8,153),(9,154),(10,155),(11,156),(12,157),(13,158),(14,159),(15,160),(16,141),(17,142),(18,143),(19,144),(20,145),(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,55),(34,56),(35,57),(36,58),(37,59),(38,60),(39,41),(40,42),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,99),(79,100),(80,81),(101,138),(102,139),(103,140),(104,121),(105,122),(106,123),(107,124),(108,125),(109,126),(110,127),(111,128),(112,129),(113,130),(114,131),(115,132),(116,133),(117,134),(118,135),(119,136),(120,137)], [(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)], [(1,64,141,90,11,74,151,100),(2,63,142,89,12,73,152,99),(3,62,143,88,13,72,153,98),(4,61,144,87,14,71,154,97),(5,80,145,86,15,70,155,96),(6,79,146,85,16,69,156,95),(7,78,147,84,17,68,157,94),(8,77,148,83,18,67,158,93),(9,76,149,82,19,66,159,92),(10,75,150,81,20,65,160,91),(21,113,48,125,31,103,58,135),(22,112,49,124,32,102,59,134),(23,111,50,123,33,101,60,133),(24,110,51,122,34,120,41,132),(25,109,52,121,35,119,42,131),(26,108,53,140,36,118,43,130),(27,107,54,139,37,117,44,129),(28,106,55,138,38,116,45,128),(29,105,56,137,39,115,46,127),(30,104,57,136,40,114,47,126)], [(1,33,6,38,11,23,16,28),(2,22,7,27,12,32,17,37),(3,31,8,36,13,21,18,26),(4,40,9,25,14,30,19,35),(5,29,10,34,15,39,20,24),(41,145,46,150,51,155,56,160),(42,154,47,159,52,144,57,149),(43,143,48,148,53,153,58,158),(44,152,49,157,54,142,59,147),(45,141,50,146,55,151,60,156),(61,131,66,136,71,121,76,126),(62,140,67,125,72,130,77,135),(63,129,68,134,73,139,78,124),(64,138,69,123,74,128,79,133),(65,127,70,132,75,137,80,122),(81,105,86,110,91,115,96,120),(82,114,87,119,92,104,97,109),(83,103,88,108,93,113,98,118),(84,112,89,117,94,102,99,107),(85,101,90,106,95,111,100,116)]])
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