metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.12D4, C4.97(D4×D5), Q8⋊C4⋊8D5, (C2×D40).3C2, (C2×C8).20D10, D20⋊8C4⋊5C2, C4⋊C4.157D10, C4.8(C4○D20), C20.129(C2×D4), C5⋊2(D4.2D4), (C2×Q8).24D10, C20.8Q8⋊8C2, D20⋊6C4⋊13C2, C10.72(C4○D8), C20.24(C4○D4), C20.23D4⋊2C2, (C2×C40).20C22, (C2×Dic5).44D4, C22.208(D4×D5), C2.19(D40⋊C2), C10.28(C4⋊D4), C10.66(C8⋊C22), (C2×C20).259C23, (C2×D20).73C22, (Q8×C10).42C22, C2.11(Q8.D10), C2.31(D10⋊D4), (C4×Dic5).32C22, (C2×Q8⋊D5)⋊6C2, (C5×Q8⋊C4)⋊8C2, (C2×C10).272(C2×D4), (C5×C4⋊C4).60C22, (C2×C5⋊2C8).49C22, (C2×C4).366(C22×D5), SmallGroup(320,446)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.12D4
G = < a,b,c,d | a20=b2=c4=1, d2=a15, bab=a-1, cac-1=a11, ad=da, bc=cb, dbd-1=a15b, dcd-1=a5c-1 >
Subgroups: 630 in 124 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×6], D4 [×5], Q8 [×2], C23 [×2], D5 [×3], C10 [×3], C42, C22⋊C4 [×3], C4⋊C4, C2×C8, C2×C8, D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×7], C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C5⋊2C8, C40, C4×D5 [×2], D20 [×2], D20 [×3], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5 [×2], D4.2D4, D40 [×2], C2×C5⋊2C8, C4×Dic5, D10⋊C4 [×3], Q8⋊D5 [×2], C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20 [×2], Q8×C10, D20⋊6C4, C20.8Q8, C5×Q8⋊C4, D20⋊8C4, C2×D40, C2×Q8⋊D5, C20.23D4, D20.12D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C4○D8, C8⋊C22, C22×D5, D4.2D4, C4○D20, D4×D5 [×2], D10⋊D4, D40⋊C2, Q8.D10, D20.12D4
►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)
(1 128)(2 127)(3 126)(4 125)(5 124)(6 123)(7 122)(8 121)(9 140)(10 139)(11 138)(12 137)(13 136)(14 135)(15 134)(16 133)(17 132)(18 131)(19 130)(20 129)(21 151)(22 150)(23 149)(24 148)(25 147)(26 146)(27 145)(28 144)(29 143)(30 142)(31 141)(32 160)(33 159)(34 158)(35 157)(36 156)(37 155)(38 154)(39 153)(40 152)(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)(61 102)(62 101)(63 120)(64 119)(65 118)(66 117)(67 116)(68 115)(69 114)(70 113)(71 112)(72 111)(73 110)(74 109)(75 108)(76 107)(77 106)(78 105)(79 104)(80 103)
(1 110 37 41)(2 101 38 52)(3 112 39 43)(4 103 40 54)(5 114 21 45)(6 105 22 56)(7 116 23 47)(8 107 24 58)(9 118 25 49)(10 109 26 60)(11 120 27 51)(12 111 28 42)(13 102 29 53)(14 113 30 44)(15 104 31 55)(16 115 32 46)(17 106 33 57)(18 117 34 48)(19 108 35 59)(20 119 36 50)(61 143 88 136)(62 154 89 127)(63 145 90 138)(64 156 91 129)(65 147 92 140)(66 158 93 131)(67 149 94 122)(68 160 95 133)(69 151 96 124)(70 142 97 135)(71 153 98 126)(72 144 99 137)(73 155 100 128)(74 146 81 139)(75 157 82 130)(76 148 83 121)(77 159 84 132)(78 150 85 123)(79 141 86 134)(80 152 87 125)
(1 146 16 141 11 156 6 151)(2 147 17 142 12 157 7 152)(3 148 18 143 13 158 8 153)(4 149 19 144 14 159 9 154)(5 150 20 145 15 160 10 155)(21 123 36 138 31 133 26 128)(22 124 37 139 32 134 27 129)(23 125 38 140 33 135 28 130)(24 126 39 121 34 136 29 131)(25 127 40 122 35 137 30 132)(41 86 56 81 51 96 46 91)(42 87 57 82 52 97 47 92)(43 88 58 83 53 98 48 93)(44 89 59 84 54 99 49 94)(45 90 60 85 55 100 50 95)(61 107 76 102 71 117 66 112)(62 108 77 103 72 118 67 113)(63 109 78 104 73 119 68 114)(64 110 79 105 74 120 69 115)(65 111 80 106 75 101 70 116)
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), (1,128)(2,127)(3,126)(4,125)(5,124)(6,123)(7,122)(8,121)(9,140)(10,139)(11,138)(12,137)(13,136)(14,135)(15,134)(16,133)(17,132)(18,131)(19,130)(20,129)(21,151)(22,150)(23,149)(24,148)(25,147)(26,146)(27,145)(28,144)(29,143)(30,142)(31,141)(32,160)(33,159)(34,158)(35,157)(36,156)(37,155)(38,154)(39,153)(40,152)(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)(61,102)(62,101)(63,120)(64,119)(65,118)(66,117)(67,116)(68,115)(69,114)(70,113)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103), (1,110,37,41)(2,101,38,52)(3,112,39,43)(4,103,40,54)(5,114,21,45)(6,105,22,56)(7,116,23,47)(8,107,24,58)(9,118,25,49)(10,109,26,60)(11,120,27,51)(12,111,28,42)(13,102,29,53)(14,113,30,44)(15,104,31,55)(16,115,32,46)(17,106,33,57)(18,117,34,48)(19,108,35,59)(20,119,36,50)(61,143,88,136)(62,154,89,127)(63,145,90,138)(64,156,91,129)(65,147,92,140)(66,158,93,131)(67,149,94,122)(68,160,95,133)(69,151,96,124)(70,142,97,135)(71,153,98,126)(72,144,99,137)(73,155,100,128)(74,146,81,139)(75,157,82,130)(76,148,83,121)(77,159,84,132)(78,150,85,123)(79,141,86,134)(80,152,87,125), (1,146,16,141,11,156,6,151)(2,147,17,142,12,157,7,152)(3,148,18,143,13,158,8,153)(4,149,19,144,14,159,9,154)(5,150,20,145,15,160,10,155)(21,123,36,138,31,133,26,128)(22,124,37,139,32,134,27,129)(23,125,38,140,33,135,28,130)(24,126,39,121,34,136,29,131)(25,127,40,122,35,137,30,132)(41,86,56,81,51,96,46,91)(42,87,57,82,52,97,47,92)(43,88,58,83,53,98,48,93)(44,89,59,84,54,99,49,94)(45,90,60,85,55,100,50,95)(61,107,76,102,71,117,66,112)(62,108,77,103,72,118,67,113)(63,109,78,104,73,119,68,114)(64,110,79,105,74,120,69,115)(65,111,80,106,75,101,70,116)>;
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), (1,128)(2,127)(3,126)(4,125)(5,124)(6,123)(7,122)(8,121)(9,140)(10,139)(11,138)(12,137)(13,136)(14,135)(15,134)(16,133)(17,132)(18,131)(19,130)(20,129)(21,151)(22,150)(23,149)(24,148)(25,147)(26,146)(27,145)(28,144)(29,143)(30,142)(31,141)(32,160)(33,159)(34,158)(35,157)(36,156)(37,155)(38,154)(39,153)(40,152)(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)(61,102)(62,101)(63,120)(64,119)(65,118)(66,117)(67,116)(68,115)(69,114)(70,113)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103), (1,110,37,41)(2,101,38,52)(3,112,39,43)(4,103,40,54)(5,114,21,45)(6,105,22,56)(7,116,23,47)(8,107,24,58)(9,118,25,49)(10,109,26,60)(11,120,27,51)(12,111,28,42)(13,102,29,53)(14,113,30,44)(15,104,31,55)(16,115,32,46)(17,106,33,57)(18,117,34,48)(19,108,35,59)(20,119,36,50)(61,143,88,136)(62,154,89,127)(63,145,90,138)(64,156,91,129)(65,147,92,140)(66,158,93,131)(67,149,94,122)(68,160,95,133)(69,151,96,124)(70,142,97,135)(71,153,98,126)(72,144,99,137)(73,155,100,128)(74,146,81,139)(75,157,82,130)(76,148,83,121)(77,159,84,132)(78,150,85,123)(79,141,86,134)(80,152,87,125), (1,146,16,141,11,156,6,151)(2,147,17,142,12,157,7,152)(3,148,18,143,13,158,8,153)(4,149,19,144,14,159,9,154)(5,150,20,145,15,160,10,155)(21,123,36,138,31,133,26,128)(22,124,37,139,32,134,27,129)(23,125,38,140,33,135,28,130)(24,126,39,121,34,136,29,131)(25,127,40,122,35,137,30,132)(41,86,56,81,51,96,46,91)(42,87,57,82,52,97,47,92)(43,88,58,83,53,98,48,93)(44,89,59,84,54,99,49,94)(45,90,60,85,55,100,50,95)(61,107,76,102,71,117,66,112)(62,108,77,103,72,118,67,113)(63,109,78,104,73,119,68,114)(64,110,79,105,74,120,69,115)(65,111,80,106,75,101,70,116) );
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)], [(1,128),(2,127),(3,126),(4,125),(5,124),(6,123),(7,122),(8,121),(9,140),(10,139),(11,138),(12,137),(13,136),(14,135),(15,134),(16,133),(17,132),(18,131),(19,130),(20,129),(21,151),(22,150),(23,149),(24,148),(25,147),(26,146),(27,145),(28,144),(29,143),(30,142),(31,141),(32,160),(33,159),(34,158),(35,157),(36,156),(37,155),(38,154),(39,153),(40,152),(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),(61,102),(62,101),(63,120),(64,119),(65,118),(66,117),(67,116),(68,115),(69,114),(70,113),(71,112),(72,111),(73,110),(74,109),(75,108),(76,107),(77,106),(78,105),(79,104),(80,103)], [(1,110,37,41),(2,101,38,52),(3,112,39,43),(4,103,40,54),(5,114,21,45),(6,105,22,56),(7,116,23,47),(8,107,24,58),(9,118,25,49),(10,109,26,60),(11,120,27,51),(12,111,28,42),(13,102,29,53),(14,113,30,44),(15,104,31,55),(16,115,32,46),(17,106,33,57),(18,117,34,48),(19,108,35,59),(20,119,36,50),(61,143,88,136),(62,154,89,127),(63,145,90,138),(64,156,91,129),(65,147,92,140),(66,158,93,131),(67,149,94,122),(68,160,95,133),(69,151,96,124),(70,142,97,135),(71,153,98,126),(72,144,99,137),(73,155,100,128),(74,146,81,139),(75,157,82,130),(76,148,83,121),(77,159,84,132),(78,150,85,123),(79,141,86,134),(80,152,87,125)], [(1,146,16,141,11,156,6,151),(2,147,17,142,12,157,7,152),(3,148,18,143,13,158,8,153),(4,149,19,144,14,159,9,154),(5,150,20,145,15,160,10,155),(21,123,36,138,31,133,26,128),(22,124,37,139,32,134,27,129),(23,125,38,140,33,135,28,130),(24,126,39,121,34,136,29,131),(25,127,40,122,35,137,30,132),(41,86,56,81,51,96,46,91),(42,87,57,82,52,97,47,92),(43,88,58,83,53,98,48,93),(44,89,59,84,54,99,49,94),(45,90,60,85,55,100,50,95),(61,107,76,102,71,117,66,112),(62,108,77,103,72,118,67,113),(63,109,78,104,73,119,68,114),(64,110,79,105,74,120,69,115),(65,111,80,106,75,101,70,116)])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 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 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4○D20 | C8⋊C22 | D4×D5 | D4×D5 | D40⋊C2 | Q8.D10 |
| kernel | D20.12D4 | D20⋊6C4 | C20.8Q8 | C5×Q8⋊C4 | D20⋊8C4 | C2×D40 | C2×Q8⋊D5 | C20.23D4 | D20 | C2×Dic5 | Q8⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | C10 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D20.12D4 ►in GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 7 | 0 | 0 |
| 0 | 0 | 40 | 2 |
| 0 | 0 | 40 | 1 |
| 32 | 30 | 0 | 0 |
| 11 | 9 | 0 | 0 |
| 0 | 0 | 17 | 24 |
| 0 | 0 | 29 | 24 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 30 | 11 |
| 0 | 0 | 15 | 11 |
| 30 | 32 | 0 | 0 |
| 9 | 11 | 0 | 0 |
| 0 | 0 | 24 | 17 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,7,0,0,0,0,40,40,0,0,2,1],[32,11,0,0,30,9,0,0,0,0,17,29,0,0,24,24],[9,0,0,0,0,9,0,0,0,0,30,15,0,0,11,11],[30,9,0,0,32,11,0,0,0,0,24,12,0,0,17,0] >;
D20.12D4 in GAP, Magma, Sage, TeX
D_{20}._{12}D_4 % in TeX
G:=Group("D20.12D4"); // GroupNames label
G:=SmallGroup(320,446);
// by ID
G=gap.SmallGroup(320,446);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,184,1684,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^15,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=a^5*c^-1>;
// generators/relations