metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2D20, D10⋊7SD16, Dic10⋊4D4, (C5×Q8)⋊1D4, C4.5(C2×D20), C4.92(D4×D5), C4⋊C4.23D10, C5⋊2(Q8⋊D4), Q8⋊C4⋊11D5, C20.120(C2×D4), C4⋊D20.2C2, (C2×C8).123D10, D10⋊1C8⋊13C2, C2.17(D5×SD16), C10.23C22≀C2, C10.Q16⋊11C2, (C2×Q8).107D10, (C2×Dic5).38D4, C10.31(C2×SD16), C22.196(D4×D5), (C2×C40).134C22, (C2×C20).246C23, (C22×D5).114D4, (C2×D20).66C22, (Q8×C10).29C22, C2.26(C22⋊D20), C2.14(Q16⋊D5), C10.60(C8.C22), (C2×Dic10).74C22, (C2×Q8×D5)⋊1C2, (C2×Q8⋊D5)⋊2C2, (C2×C40⋊C2)⋊17C2, (C2×C4×D5).24C22, (C5×Q8⋊C4)⋊11C2, (C2×C10).259(C2×D4), (C5×C4⋊C4).47C22, (C2×C5⋊2C8).38C22, (C2×C4).353(C22×D5), SmallGroup(320,433)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2D20
G = < a,b,c,d | a4=c20=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=dbd=ab, dcd=c-1 >
Subgroups: 750 in 158 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C22⋊C8, Q8⋊C4, Q8⋊C4, C4⋊D4, C2×SD16, C22×Q8, C5⋊2C8, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C22×D5, Q8⋊D4, C40⋊C2, C2×C5⋊2C8, D10⋊C4, Q8⋊D5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8×D5, Q8×C10, C10.Q16, D10⋊1C8, C5×Q8⋊C4, C4⋊D20, C2×C40⋊C2, C2×Q8⋊D5, C2×Q8×D5, Q8⋊2D20
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C22≀C2, C2×SD16, C8.C22, D20, C22×D5, Q8⋊D4, C2×D20, D4×D5, C22⋊D20, D5×SD16, Q16⋊D5, Q8⋊2D20
►On 160 points
Generators in S160
(1 38 96 138)(2 139 97 39)(3 40 98 140)(4 121 99 21)(5 22 100 122)(6 123 81 23)(7 24 82 124)(8 125 83 25)(9 26 84 126)(10 127 85 27)(11 28 86 128)(12 129 87 29)(13 30 88 130)(14 131 89 31)(15 32 90 132)(16 133 91 33)(17 34 92 134)(18 135 93 35)(19 36 94 136)(20 137 95 37)(41 61 115 154)(42 155 116 62)(43 63 117 156)(44 157 118 64)(45 65 119 158)(46 159 120 66)(47 67 101 160)(48 141 102 68)(49 69 103 142)(50 143 104 70)(51 71 105 144)(52 145 106 72)(53 73 107 146)(54 147 108 74)(55 75 109 148)(56 149 110 76)(57 77 111 150)(58 151 112 78)(59 79 113 152)(60 153 114 80)
(1 109 96 55)(2 76 97 149)(3 111 98 57)(4 78 99 151)(5 113 100 59)(6 80 81 153)(7 115 82 41)(8 62 83 155)(9 117 84 43)(10 64 85 157)(11 119 86 45)(12 66 87 159)(13 101 88 47)(14 68 89 141)(15 103 90 49)(16 70 91 143)(17 105 92 51)(18 72 93 145)(19 107 94 53)(20 74 95 147)(21 58 121 112)(22 79 122 152)(23 60 123 114)(24 61 124 154)(25 42 125 116)(26 63 126 156)(27 44 127 118)(28 65 128 158)(29 46 129 120)(30 67 130 160)(31 48 131 102)(32 69 132 142)(33 50 133 104)(34 71 134 144)(35 52 135 106)(36 73 136 146)(37 54 137 108)(38 75 138 148)(39 56 139 110)(40 77 140 150)
(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)(21 129)(22 128)(23 127)(24 126)(25 125)(26 124)(27 123)(28 122)(29 121)(30 140)(31 139)(32 138)(33 137)(34 136)(35 135)(36 134)(37 133)(38 132)(39 131)(40 130)(41 156)(42 155)(43 154)(44 153)(45 152)(46 151)(47 150)(48 149)(49 148)(50 147)(51 146)(52 145)(53 144)(54 143)(55 142)(56 141)(57 160)(58 159)(59 158)(60 157)(61 117)(62 116)(63 115)(64 114)(65 113)(66 112)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 120)(79 119)(80 118)(81 85)(82 84)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)
G:=sub<Sym(160)| (1,38,96,138)(2,139,97,39)(3,40,98,140)(4,121,99,21)(5,22,100,122)(6,123,81,23)(7,24,82,124)(8,125,83,25)(9,26,84,126)(10,127,85,27)(11,28,86,128)(12,129,87,29)(13,30,88,130)(14,131,89,31)(15,32,90,132)(16,133,91,33)(17,34,92,134)(18,135,93,35)(19,36,94,136)(20,137,95,37)(41,61,115,154)(42,155,116,62)(43,63,117,156)(44,157,118,64)(45,65,119,158)(46,159,120,66)(47,67,101,160)(48,141,102,68)(49,69,103,142)(50,143,104,70)(51,71,105,144)(52,145,106,72)(53,73,107,146)(54,147,108,74)(55,75,109,148)(56,149,110,76)(57,77,111,150)(58,151,112,78)(59,79,113,152)(60,153,114,80), (1,109,96,55)(2,76,97,149)(3,111,98,57)(4,78,99,151)(5,113,100,59)(6,80,81,153)(7,115,82,41)(8,62,83,155)(9,117,84,43)(10,64,85,157)(11,119,86,45)(12,66,87,159)(13,101,88,47)(14,68,89,141)(15,103,90,49)(16,70,91,143)(17,105,92,51)(18,72,93,145)(19,107,94,53)(20,74,95,147)(21,58,121,112)(22,79,122,152)(23,60,123,114)(24,61,124,154)(25,42,125,116)(26,63,126,156)(27,44,127,118)(28,65,128,158)(29,46,129,120)(30,67,130,160)(31,48,131,102)(32,69,132,142)(33,50,133,104)(34,71,134,144)(35,52,135,106)(36,73,136,146)(37,54,137,108)(38,75,138,148)(39,56,139,110)(40,77,140,150), (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)(21,129)(22,128)(23,127)(24,126)(25,125)(26,124)(27,123)(28,122)(29,121)(30,140)(31,139)(32,138)(33,137)(34,136)(35,135)(36,134)(37,133)(38,132)(39,131)(40,130)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,160)(58,159)(59,158)(60,157)(61,117)(62,116)(63,115)(64,114)(65,113)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,120)(79,119)(80,118)(81,85)(82,84)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)>;
G:=Group( (1,38,96,138)(2,139,97,39)(3,40,98,140)(4,121,99,21)(5,22,100,122)(6,123,81,23)(7,24,82,124)(8,125,83,25)(9,26,84,126)(10,127,85,27)(11,28,86,128)(12,129,87,29)(13,30,88,130)(14,131,89,31)(15,32,90,132)(16,133,91,33)(17,34,92,134)(18,135,93,35)(19,36,94,136)(20,137,95,37)(41,61,115,154)(42,155,116,62)(43,63,117,156)(44,157,118,64)(45,65,119,158)(46,159,120,66)(47,67,101,160)(48,141,102,68)(49,69,103,142)(50,143,104,70)(51,71,105,144)(52,145,106,72)(53,73,107,146)(54,147,108,74)(55,75,109,148)(56,149,110,76)(57,77,111,150)(58,151,112,78)(59,79,113,152)(60,153,114,80), (1,109,96,55)(2,76,97,149)(3,111,98,57)(4,78,99,151)(5,113,100,59)(6,80,81,153)(7,115,82,41)(8,62,83,155)(9,117,84,43)(10,64,85,157)(11,119,86,45)(12,66,87,159)(13,101,88,47)(14,68,89,141)(15,103,90,49)(16,70,91,143)(17,105,92,51)(18,72,93,145)(19,107,94,53)(20,74,95,147)(21,58,121,112)(22,79,122,152)(23,60,123,114)(24,61,124,154)(25,42,125,116)(26,63,126,156)(27,44,127,118)(28,65,128,158)(29,46,129,120)(30,67,130,160)(31,48,131,102)(32,69,132,142)(33,50,133,104)(34,71,134,144)(35,52,135,106)(36,73,136,146)(37,54,137,108)(38,75,138,148)(39,56,139,110)(40,77,140,150), (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)(21,129)(22,128)(23,127)(24,126)(25,125)(26,124)(27,123)(28,122)(29,121)(30,140)(31,139)(32,138)(33,137)(34,136)(35,135)(36,134)(37,133)(38,132)(39,131)(40,130)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,160)(58,159)(59,158)(60,157)(61,117)(62,116)(63,115)(64,114)(65,113)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,120)(79,119)(80,118)(81,85)(82,84)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94) );
G=PermutationGroup([[(1,38,96,138),(2,139,97,39),(3,40,98,140),(4,121,99,21),(5,22,100,122),(6,123,81,23),(7,24,82,124),(8,125,83,25),(9,26,84,126),(10,127,85,27),(11,28,86,128),(12,129,87,29),(13,30,88,130),(14,131,89,31),(15,32,90,132),(16,133,91,33),(17,34,92,134),(18,135,93,35),(19,36,94,136),(20,137,95,37),(41,61,115,154),(42,155,116,62),(43,63,117,156),(44,157,118,64),(45,65,119,158),(46,159,120,66),(47,67,101,160),(48,141,102,68),(49,69,103,142),(50,143,104,70),(51,71,105,144),(52,145,106,72),(53,73,107,146),(54,147,108,74),(55,75,109,148),(56,149,110,76),(57,77,111,150),(58,151,112,78),(59,79,113,152),(60,153,114,80)], [(1,109,96,55),(2,76,97,149),(3,111,98,57),(4,78,99,151),(5,113,100,59),(6,80,81,153),(7,115,82,41),(8,62,83,155),(9,117,84,43),(10,64,85,157),(11,119,86,45),(12,66,87,159),(13,101,88,47),(14,68,89,141),(15,103,90,49),(16,70,91,143),(17,105,92,51),(18,72,93,145),(19,107,94,53),(20,74,95,147),(21,58,121,112),(22,79,122,152),(23,60,123,114),(24,61,124,154),(25,42,125,116),(26,63,126,156),(27,44,127,118),(28,65,128,158),(29,46,129,120),(30,67,130,160),(31,48,131,102),(32,69,132,142),(33,50,133,104),(34,71,134,144),(35,52,135,106),(36,73,136,146),(37,54,137,108),(38,75,138,148),(39,56,139,110),(40,77,140,150)], [(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),(21,129),(22,128),(23,127),(24,126),(25,125),(26,124),(27,123),(28,122),(29,121),(30,140),(31,139),(32,138),(33,137),(34,136),(35,135),(36,134),(37,133),(38,132),(39,131),(40,130),(41,156),(42,155),(43,154),(44,153),(45,152),(46,151),(47,150),(48,149),(49,148),(50,147),(51,146),(52,145),(53,144),(54,143),(55,142),(56,141),(57,160),(58,159),(59,158),(60,157),(61,117),(62,116),(63,115),(64,114),(65,113),(66,112),(67,111),(68,110),(69,109),(70,108),(71,107),(72,106),(73,105),(74,104),(75,103),(76,102),(77,101),(78,120),(79,119),(80,118),(81,85),(82,84),(86,100),(87,99),(88,98),(89,97),(90,96),(91,95),(92,94)]])
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 | 10 | 10 | 40 | 2 | 2 | 4 | 4 | 8 | 20 | 20 | 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 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | SD16 | D10 | D10 | D10 | D20 | C8.C22 | D4×D5 | D4×D5 | D5×SD16 | Q16⋊D5 |
| kernel | Q8⋊2D20 | C10.Q16 | D10⋊1C8 | C5×Q8⋊C4 | C4⋊D20 | C2×C40⋊C2 | C2×Q8⋊D5 | C2×Q8×D5 | Dic10 | C2×Dic5 | C5×Q8 | C22×D5 | Q8⋊C4 | D10 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Q8⋊2D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 15 | 26 |
| 0 | 0 | 26 | 26 |
| 30 | 39 | 0 | 0 |
| 16 | 14 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 8 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,15,26,0,0,26,26],[30,16,0,0,39,14,0,0,0,0,1,0,0,0,0,40],[1,8,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;
Q8⋊2D20 in GAP, Magma, Sage, TeX
Q_8\rtimes_2D_{20} % in TeX
G:=Group("Q8:2D20"); // GroupNames label
G:=SmallGroup(320,433);
// by ID
G=gap.SmallGroup(320,433);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,58,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^20=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=a*b,d*c*d=c^-1>;
// generators/relations