metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊8D20, C40⋊13D4, D10⋊3SD16, C4.Q8⋊8D5, C5⋊3(C8⋊8D4), C4⋊C4.38D10, C4.50(C2×D20), D10⋊2Q8⋊6C2, (C2×C8).260D10, C4⋊D20.6C2, C20.130(C2×D4), D20⋊6C4⋊15C2, C2.24(D5×SD16), C10.55(C4○D8), C20.29(C4○D4), C10.Q16⋊16C2, C4.3(Q8⋊2D5), C10.40(C2×SD16), (C22×D5).84D4, C22.216(D4×D5), C10.43(C4⋊D4), C2.16(C4⋊D20), (C2×C40).161C22, (C2×C20).280C23, (C2×Dic5).144D4, (C2×D20).78C22, C2.22(SD16⋊3D5), (C2×Dic10).87C22, (D5×C2×C8)⋊7C2, (C5×C4.Q8)⋊9C2, (C2×C40⋊C2)⋊28C2, (C2×C10).285(C2×D4), (C5×C4⋊C4).73C22, (C2×C4×D5).303C22, (C2×C4).383(C22×D5), (C2×C5⋊2C8).236C22, SmallGroup(320,491)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊8D20
G = < a,b,c | a8=b20=c2=1, bab-1=cac=a3, cbc=b-1 >
Subgroups: 598 in 124 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C8⋊8D4, C8×D5, C40⋊C2, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C2×D20, D20⋊6C4, C10.Q16, C5×C4.Q8, C4⋊D20, D10⋊2Q8, D5×C2×C8, C2×C40⋊C2, C8⋊8D20
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C4○D8, D20, C22×D5, C8⋊8D4, C2×D20, D4×D5, Q8⋊2D5, C4⋊D20, D5×SD16, SD16⋊3D5, C8⋊8D20
►On 160 points
Generators in S160
(1 104 49 149 62 140 39 91)(2 150 40 105 63 92 50 121)(3 106 51 151 64 122 21 93)(4 152 22 107 65 94 52 123)(5 108 53 153 66 124 23 95)(6 154 24 109 67 96 54 125)(7 110 55 155 68 126 25 97)(8 156 26 111 69 98 56 127)(9 112 57 157 70 128 27 99)(10 158 28 113 71 100 58 129)(11 114 59 159 72 130 29 81)(12 160 30 115 73 82 60 131)(13 116 41 141 74 132 31 83)(14 142 32 117 75 84 42 133)(15 118 43 143 76 134 33 85)(16 144 34 119 77 86 44 135)(17 120 45 145 78 136 35 87)(18 146 36 101 79 88 46 137)(19 102 47 147 80 138 37 89)(20 148 38 103 61 90 48 139)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 41)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 54)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(78 80)(81 124)(82 123)(83 122)(84 121)(85 140)(86 139)(87 138)(88 137)(89 136)(90 135)(91 134)(92 133)(93 132)(94 131)(95 130)(96 129)(97 128)(98 127)(99 126)(100 125)(101 146)(102 145)(103 144)(104 143)(105 142)(106 141)(107 160)(108 159)(109 158)(110 157)(111 156)(112 155)(113 154)(114 153)(115 152)(116 151)(117 150)(118 149)(119 148)(120 147)
G:=sub<Sym(160)| (1,104,49,149,62,140,39,91)(2,150,40,105,63,92,50,121)(3,106,51,151,64,122,21,93)(4,152,22,107,65,94,52,123)(5,108,53,153,66,124,23,95)(6,154,24,109,67,96,54,125)(7,110,55,155,68,126,25,97)(8,156,26,111,69,98,56,127)(9,112,57,157,70,128,27,99)(10,158,28,113,71,100,58,129)(11,114,59,159,72,130,29,81)(12,160,30,115,73,82,60,131)(13,116,41,141,74,132,31,83)(14,142,32,117,75,84,42,133)(15,118,43,143,76,134,33,85)(16,144,34,119,77,86,44,135)(17,120,45,145,78,136,35,87)(18,146,36,101,79,88,46,137)(19,102,47,147,80,138,37,89)(20,148,38,103,61,90,48,139), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,41)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,80)(81,124)(82,123)(83,122)(84,121)(85,140)(86,139)(87,138)(88,137)(89,136)(90,135)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129)(97,128)(98,127)(99,126)(100,125)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,160)(108,159)(109,158)(110,157)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147)>;
G:=Group( (1,104,49,149,62,140,39,91)(2,150,40,105,63,92,50,121)(3,106,51,151,64,122,21,93)(4,152,22,107,65,94,52,123)(5,108,53,153,66,124,23,95)(6,154,24,109,67,96,54,125)(7,110,55,155,68,126,25,97)(8,156,26,111,69,98,56,127)(9,112,57,157,70,128,27,99)(10,158,28,113,71,100,58,129)(11,114,59,159,72,130,29,81)(12,160,30,115,73,82,60,131)(13,116,41,141,74,132,31,83)(14,142,32,117,75,84,42,133)(15,118,43,143,76,134,33,85)(16,144,34,119,77,86,44,135)(17,120,45,145,78,136,35,87)(18,146,36,101,79,88,46,137)(19,102,47,147,80,138,37,89)(20,148,38,103,61,90,48,139), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,41)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,80)(81,124)(82,123)(83,122)(84,121)(85,140)(86,139)(87,138)(88,137)(89,136)(90,135)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129)(97,128)(98,127)(99,126)(100,125)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,160)(108,159)(109,158)(110,157)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147) );
G=PermutationGroup([[(1,104,49,149,62,140,39,91),(2,150,40,105,63,92,50,121),(3,106,51,151,64,122,21,93),(4,152,22,107,65,94,52,123),(5,108,53,153,66,124,23,95),(6,154,24,109,67,96,54,125),(7,110,55,155,68,126,25,97),(8,156,26,111,69,98,56,127),(9,112,57,157,70,128,27,99),(10,158,28,113,71,100,58,129),(11,114,59,159,72,130,29,81),(12,160,30,115,73,82,60,131),(13,116,41,141,74,132,31,83),(14,142,32,117,75,84,42,133),(15,118,43,143,76,134,33,85),(16,144,34,119,77,86,44,135),(17,120,45,145,78,136,35,87),(18,146,36,101,79,88,46,137),(19,102,47,147,80,138,37,89),(20,148,38,103,61,90,48,139)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,41),(22,60),(23,59),(24,58),(25,57),(26,56),(27,55),(28,54),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(78,80),(81,124),(82,123),(83,122),(84,121),(85,140),(86,139),(87,138),(88,137),(89,136),(90,135),(91,134),(92,133),(93,132),(94,131),(95,130),(96,129),(97,128),(98,127),(99,126),(100,125),(101,146),(102,145),(103,144),(104,143),(105,142),(106,141),(107,160),(108,159),(109,158),(110,157),(111,156),(112,155),(113,154),(114,153),(115,152),(116,151),(117,150),(118,149),(119,148),(120,147)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 40 | 2 | 2 | 8 | 8 | 10 | 10 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | SD16 | D10 | D10 | C4○D8 | D20 | Q8⋊2D5 | D4×D5 | D5×SD16 | SD16⋊3D5 |
| kernel | C8⋊8D20 | D20⋊6C4 | C10.Q16 | C5×C4.Q8 | C4⋊D20 | D10⋊2Q8 | D5×C2×C8 | C2×C40⋊C2 | C40 | C2×Dic5 | C22×D5 | C4.Q8 | C20 | D10 | C4⋊C4 | C2×C8 | C10 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of C8⋊8D20 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 30 |
| 0 | 0 | 0 | 0 | 26 | 30 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 5 | 0 | 0 |
| 0 | 0 | 16 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 25 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,26,0,0,0,0,30,30],[0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,16,0,0,0,0,5,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,25,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;
C8⋊8D20 in GAP, Magma, Sage, TeX
C_8\rtimes_8D_{20} % in TeX
G:=Group("C8:8D20"); // GroupNames label
G:=SmallGroup(320,491);
// by ID
G=gap.SmallGroup(320,491);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,555,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^8=b^20=c^2=1,b*a*b^-1=c*a*c=a^3,c*b*c=b^-1>;
// generators/relations