metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊7Q8, C42.149D10, C10.1312+ (1+4), C20⋊Q8⋊36C2, C4.16(Q8×D5), C5⋊7(D4⋊3Q8), C20.51(C2×Q8), C42.C2⋊5D5, C4⋊C4.205D10, (C4×D20).24C2, D10.23(C2×Q8), D10⋊2Q8⋊35C2, D10⋊Q8⋊34C2, (C4×Dic10)⋊47C2, (C2×C20).88C23, C4.Dic10⋊34C2, D20⋊8C4.11C2, C10.43(C22×Q8), (C2×C10).234C24, (C4×C20).194C22, C2.56(D4⋊8D10), Dic5.45(C4○D4), (C2×D20).275C22, C4⋊Dic5.379C22, C22.255(C23×D5), D10⋊C4.40C22, (C4×Dic5).149C22, (C2×Dic5).122C23, (C22×D5).231C23, (C2×Dic10).259C22, C10.D4.144C22, (D5×C4⋊C4)⋊35C2, C2.26(C2×Q8×D5), C2.85(D5×C4○D4), (C5×C42.C2)⋊7C2, C10.196(C2×C4○D4), (C2×C4×D5).268C22, (C2×C4).78(C22×D5), (C5×C4⋊C4).189C22, SmallGroup(320,1362)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 806 in 228 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×13], C22, C22 [×8], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×4], Q8 [×4], C23 [×2], D5 [×4], C10 [×3], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C22×C4 [×6], C2×D4, C2×Q8 [×3], Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×6], D10 [×4], D10 [×4], C2×C10, C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C42.C2, C42.C2, C4⋊Q8, Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×4], C22×D5 [×2], D4⋊3Q8, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×4], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×6], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×6], C2×D20, C4×Dic10, C4×D20, C20⋊Q8, C4.Dic10, D5×C4⋊C4 [×2], D20⋊8C4 [×2], D10⋊Q8 [×4], D10⋊2Q8 [×2], C5×C42.C2, D20⋊7Q8
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), C22×D5 [×7], D4⋊3Q8, Q8×D5 [×2], C23×D5, C2×Q8×D5, D5×C4○D4, D4⋊8D10, D20⋊7Q8
Generators and relations
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a9, cbc-1=a10b, dbd-1=a18b, dcd-1=c-1 >
►On 160 points
Generators in S160
(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)
(1 113)(2 112)(3 111)(4 110)(5 109)(6 108)(7 107)(8 106)(9 105)(10 104)(11 103)(12 102)(13 101)(14 120)(15 119)(16 118)(17 117)(18 116)(19 115)(20 114)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 80)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 72)(37 71)(38 70)(39 69)(40 68)(41 100)(42 99)(43 98)(44 97)(45 96)(46 95)(47 94)(48 93)(49 92)(50 91)(51 90)(52 89)(53 88)(54 87)(55 86)(56 85)(57 84)(58 83)(59 82)(60 81)(121 155)(122 154)(123 153)(124 152)(125 151)(126 150)(127 149)(128 148)(129 147)(130 146)(131 145)(132 144)(133 143)(134 142)(135 141)(136 160)(137 159)(138 158)(139 157)(140 156)
(1 44 109 83)(2 45 110 84)(3 46 111 85)(4 47 112 86)(5 48 113 87)(6 49 114 88)(7 50 115 89)(8 51 116 90)(9 52 117 91)(10 53 118 92)(11 54 119 93)(12 55 120 94)(13 56 101 95)(14 57 102 96)(15 58 103 97)(16 59 104 98)(17 60 105 99)(18 41 106 100)(19 42 107 81)(20 43 108 82)(21 126 65 158)(22 127 66 159)(23 128 67 160)(24 129 68 141)(25 130 69 142)(26 131 70 143)(27 132 71 144)(28 133 72 145)(29 134 73 146)(30 135 74 147)(31 136 75 148)(32 137 76 149)(33 138 77 150)(34 139 78 151)(35 140 79 152)(36 121 80 153)(37 122 61 154)(38 123 62 155)(39 124 63 156)(40 125 64 157)
(1 135 109 147)(2 124 110 156)(3 133 111 145)(4 122 112 154)(5 131 113 143)(6 140 114 152)(7 129 115 141)(8 138 116 150)(9 127 117 159)(10 136 118 148)(11 125 119 157)(12 134 120 146)(13 123 101 155)(14 132 102 144)(15 121 103 153)(16 130 104 142)(17 139 105 151)(18 128 106 160)(19 137 107 149)(20 126 108 158)(21 82 65 43)(22 91 66 52)(23 100 67 41)(24 89 68 50)(25 98 69 59)(26 87 70 48)(27 96 71 57)(28 85 72 46)(29 94 73 55)(30 83 74 44)(31 92 75 53)(32 81 76 42)(33 90 77 51)(34 99 78 60)(35 88 79 49)(36 97 80 58)(37 86 61 47)(38 95 62 56)(39 84 63 45)(40 93 64 54)
G:=sub<Sym(160)| (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), (1,113)(2,112)(3,111)(4,110)(5,109)(6,108)(7,107)(8,106)(9,105)(10,104)(11,103)(12,102)(13,101)(14,120)(15,119)(16,118)(17,117)(18,116)(19,115)(20,114)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(121,155)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,160)(137,159)(138,158)(139,157)(140,156), (1,44,109,83)(2,45,110,84)(3,46,111,85)(4,47,112,86)(5,48,113,87)(6,49,114,88)(7,50,115,89)(8,51,116,90)(9,52,117,91)(10,53,118,92)(11,54,119,93)(12,55,120,94)(13,56,101,95)(14,57,102,96)(15,58,103,97)(16,59,104,98)(17,60,105,99)(18,41,106,100)(19,42,107,81)(20,43,108,82)(21,126,65,158)(22,127,66,159)(23,128,67,160)(24,129,68,141)(25,130,69,142)(26,131,70,143)(27,132,71,144)(28,133,72,145)(29,134,73,146)(30,135,74,147)(31,136,75,148)(32,137,76,149)(33,138,77,150)(34,139,78,151)(35,140,79,152)(36,121,80,153)(37,122,61,154)(38,123,62,155)(39,124,63,156)(40,125,64,157), (1,135,109,147)(2,124,110,156)(3,133,111,145)(4,122,112,154)(5,131,113,143)(6,140,114,152)(7,129,115,141)(8,138,116,150)(9,127,117,159)(10,136,118,148)(11,125,119,157)(12,134,120,146)(13,123,101,155)(14,132,102,144)(15,121,103,153)(16,130,104,142)(17,139,105,151)(18,128,106,160)(19,137,107,149)(20,126,108,158)(21,82,65,43)(22,91,66,52)(23,100,67,41)(24,89,68,50)(25,98,69,59)(26,87,70,48)(27,96,71,57)(28,85,72,46)(29,94,73,55)(30,83,74,44)(31,92,75,53)(32,81,76,42)(33,90,77,51)(34,99,78,60)(35,88,79,49)(36,97,80,58)(37,86,61,47)(38,95,62,56)(39,84,63,45)(40,93,64,54)>;
G:=Group( (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), (1,113)(2,112)(3,111)(4,110)(5,109)(6,108)(7,107)(8,106)(9,105)(10,104)(11,103)(12,102)(13,101)(14,120)(15,119)(16,118)(17,117)(18,116)(19,115)(20,114)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(121,155)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,160)(137,159)(138,158)(139,157)(140,156), (1,44,109,83)(2,45,110,84)(3,46,111,85)(4,47,112,86)(5,48,113,87)(6,49,114,88)(7,50,115,89)(8,51,116,90)(9,52,117,91)(10,53,118,92)(11,54,119,93)(12,55,120,94)(13,56,101,95)(14,57,102,96)(15,58,103,97)(16,59,104,98)(17,60,105,99)(18,41,106,100)(19,42,107,81)(20,43,108,82)(21,126,65,158)(22,127,66,159)(23,128,67,160)(24,129,68,141)(25,130,69,142)(26,131,70,143)(27,132,71,144)(28,133,72,145)(29,134,73,146)(30,135,74,147)(31,136,75,148)(32,137,76,149)(33,138,77,150)(34,139,78,151)(35,140,79,152)(36,121,80,153)(37,122,61,154)(38,123,62,155)(39,124,63,156)(40,125,64,157), (1,135,109,147)(2,124,110,156)(3,133,111,145)(4,122,112,154)(5,131,113,143)(6,140,114,152)(7,129,115,141)(8,138,116,150)(9,127,117,159)(10,136,118,148)(11,125,119,157)(12,134,120,146)(13,123,101,155)(14,132,102,144)(15,121,103,153)(16,130,104,142)(17,139,105,151)(18,128,106,160)(19,137,107,149)(20,126,108,158)(21,82,65,43)(22,91,66,52)(23,100,67,41)(24,89,68,50)(25,98,69,59)(26,87,70,48)(27,96,71,57)(28,85,72,46)(29,94,73,55)(30,83,74,44)(31,92,75,53)(32,81,76,42)(33,90,77,51)(34,99,78,60)(35,88,79,49)(36,97,80,58)(37,86,61,47)(38,95,62,56)(39,84,63,45)(40,93,64,54) );
G=PermutationGroup([(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)], [(1,113),(2,112),(3,111),(4,110),(5,109),(6,108),(7,107),(8,106),(9,105),(10,104),(11,103),(12,102),(13,101),(14,120),(15,119),(16,118),(17,117),(18,116),(19,115),(20,114),(21,67),(22,66),(23,65),(24,64),(25,63),(26,62),(27,61),(28,80),(29,79),(30,78),(31,77),(32,76),(33,75),(34,74),(35,73),(36,72),(37,71),(38,70),(39,69),(40,68),(41,100),(42,99),(43,98),(44,97),(45,96),(46,95),(47,94),(48,93),(49,92),(50,91),(51,90),(52,89),(53,88),(54,87),(55,86),(56,85),(57,84),(58,83),(59,82),(60,81),(121,155),(122,154),(123,153),(124,152),(125,151),(126,150),(127,149),(128,148),(129,147),(130,146),(131,145),(132,144),(133,143),(134,142),(135,141),(136,160),(137,159),(138,158),(139,157),(140,156)], [(1,44,109,83),(2,45,110,84),(3,46,111,85),(4,47,112,86),(5,48,113,87),(6,49,114,88),(7,50,115,89),(8,51,116,90),(9,52,117,91),(10,53,118,92),(11,54,119,93),(12,55,120,94),(13,56,101,95),(14,57,102,96),(15,58,103,97),(16,59,104,98),(17,60,105,99),(18,41,106,100),(19,42,107,81),(20,43,108,82),(21,126,65,158),(22,127,66,159),(23,128,67,160),(24,129,68,141),(25,130,69,142),(26,131,70,143),(27,132,71,144),(28,133,72,145),(29,134,73,146),(30,135,74,147),(31,136,75,148),(32,137,76,149),(33,138,77,150),(34,139,78,151),(35,140,79,152),(36,121,80,153),(37,122,61,154),(38,123,62,155),(39,124,63,156),(40,125,64,157)], [(1,135,109,147),(2,124,110,156),(3,133,111,145),(4,122,112,154),(5,131,113,143),(6,140,114,152),(7,129,115,141),(8,138,116,150),(9,127,117,159),(10,136,118,148),(11,125,119,157),(12,134,120,146),(13,123,101,155),(14,132,102,144),(15,121,103,153),(16,130,104,142),(17,139,105,151),(18,128,106,160),(19,137,107,149),(20,126,108,158),(21,82,65,43),(22,91,66,52),(23,100,67,41),(24,89,68,50),(25,98,69,59),(26,87,70,48),(27,96,71,57),(28,85,72,46),(29,94,73,55),(30,83,74,44),(31,92,75,53),(32,81,76,42),(33,90,77,51),(34,99,78,60),(35,88,79,49),(36,97,80,58),(37,86,61,47),(38,95,62,56),(39,84,63,45),(40,93,64,54)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 34 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 34 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 40 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 13 |
| 0 | 0 | 0 | 0 | 34 | 34 |
G:=sub<GL(6,GF(41))| [34,40,0,0,0,0,1,0,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,34,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[34,40,0,0,0,0,7,7,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,7,34,0,0,0,0,13,34] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | 2+ (1+4) | Q8×D5 | D5×C4○D4 | D4⋊8D10 |
| kernel | D20⋊7Q8 | C4×Dic10 | C4×D20 | C20⋊Q8 | C4.Dic10 | D5×C4⋊C4 | D20⋊8C4 | D10⋊Q8 | D10⋊2Q8 | C5×C42.C2 | D20 | C42.C2 | Dic5 | C42 | C4⋊C4 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 4 | 2 | 4 | 2 | 12 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_7Q_8 % in TeX
G:=Group("D20:7Q8"); // GroupNames label
G:=SmallGroup(320,1362);
// by ID
G=gap.SmallGroup(320,1362);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,219,184,1571,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^9,c*b*c^-1=a^10*b,d*b*d^-1=a^18*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀