metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊8Q8, Dic10⋊12D4, C42.173D10, C10.362- 1+4, C5⋊4(D4×Q8), C4⋊1(Q8×D5), C4⋊Q8⋊11D5, C20⋊3(C2×Q8), C20⋊Q8⋊44C2, D10⋊7(C2×Q8), C4.74(D4×D5), C20.72(C2×D4), C4⋊C4.218D10, (C4×D20).26C2, D10⋊Q8⋊48C2, D10⋊3Q8⋊36C2, (C4×Dic10)⋊52C2, (C2×Q8).146D10, Dic5.54(C2×D4), D20⋊8C4.13C2, C10.47(C22×Q8), (C2×C20).104C23, (C4×C20).212C22, (C2×C10).271C24, C10.101(C22×D4), (C2×D20).280C22, C4⋊Dic5.385C22, (Q8×C10).138C22, C22.292(C23×D5), (C4×Dic5).168C22, (C2×Dic5).142C23, (C22×D5).242C23, D10⋊C4.152C22, C2.37(Q8.10D10), (C2×Dic10).195C22, C10.D4.166C22, (C2×Q8×D5)⋊13C2, C2.74(C2×D4×D5), C2.30(C2×Q8×D5), (C5×C4⋊Q8)⋊13C2, (C2×C4×D5).154C22, (C5×C4⋊C4).214C22, (C2×C4).218(C22×D5), SmallGroup(320,1399)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊8Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=c-1 >
Subgroups: 966 in 280 conjugacy classes, 115 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C4⋊Q8, C22×Q8, Dic10, Dic10, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, D4×Q8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×C4×D5, C2×D20, Q8×D5, Q8×C10, C4×Dic10, C4×D20, C20⋊Q8, D20⋊8C4, D10⋊Q8, D10⋊3Q8, C5×C4⋊Q8, C2×Q8×D5, D20⋊8Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C24, D10, C22×D4, C22×Q8, 2- 1+4, C22×D5, D4×Q8, D4×D5, Q8×D5, C23×D5, C2×D4×D5, C2×Q8×D5, Q8.10D10, D20⋊8Q8
►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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(54 60)(55 59)(56 58)(61 65)(62 64)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(81 87)(82 86)(83 85)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(101 109)(102 108)(103 107)(104 106)(110 120)(111 119)(112 118)(113 117)(114 116)(121 127)(122 126)(123 125)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(156 160)(157 159)
(1 76 40 50)(2 77 21 51)(3 78 22 52)(4 79 23 53)(5 80 24 54)(6 61 25 55)(7 62 26 56)(8 63 27 57)(9 64 28 58)(10 65 29 59)(11 66 30 60)(12 67 31 41)(13 68 32 42)(14 69 33 43)(15 70 34 44)(16 71 35 45)(17 72 36 46)(18 73 37 47)(19 74 38 48)(20 75 39 49)(81 131 145 102)(82 132 146 103)(83 133 147 104)(84 134 148 105)(85 135 149 106)(86 136 150 107)(87 137 151 108)(88 138 152 109)(89 139 153 110)(90 140 154 111)(91 121 155 112)(92 122 156 113)(93 123 157 114)(94 124 158 115)(95 125 159 116)(96 126 160 117)(97 127 141 118)(98 128 142 119)(99 129 143 120)(100 130 144 101)
(1 113 40 122)(2 104 21 133)(3 115 22 124)(4 106 23 135)(5 117 24 126)(6 108 25 137)(7 119 26 128)(8 110 27 139)(9 101 28 130)(10 112 29 121)(11 103 30 132)(12 114 31 123)(13 105 32 134)(14 116 33 125)(15 107 34 136)(16 118 35 127)(17 109 36 138)(18 120 37 129)(19 111 38 140)(20 102 39 131)(41 93 67 157)(42 84 68 148)(43 95 69 159)(44 86 70 150)(45 97 71 141)(46 88 72 152)(47 99 73 143)(48 90 74 154)(49 81 75 145)(50 92 76 156)(51 83 77 147)(52 94 78 158)(53 85 79 149)(54 96 80 160)(55 87 61 151)(56 98 62 142)(57 89 63 153)(58 100 64 144)(59 91 65 155)(60 82 66 146)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,65)(62,64)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(81,87)(82,86)(83,85)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(101,109)(102,108)(103,107)(104,106)(110,120)(111,119)(112,118)(113,117)(114,116)(121,127)(122,126)(123,125)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159), (1,76,40,50)(2,77,21,51)(3,78,22,52)(4,79,23,53)(5,80,24,54)(6,61,25,55)(7,62,26,56)(8,63,27,57)(9,64,28,58)(10,65,29,59)(11,66,30,60)(12,67,31,41)(13,68,32,42)(14,69,33,43)(15,70,34,44)(16,71,35,45)(17,72,36,46)(18,73,37,47)(19,74,38,48)(20,75,39,49)(81,131,145,102)(82,132,146,103)(83,133,147,104)(84,134,148,105)(85,135,149,106)(86,136,150,107)(87,137,151,108)(88,138,152,109)(89,139,153,110)(90,140,154,111)(91,121,155,112)(92,122,156,113)(93,123,157,114)(94,124,158,115)(95,125,159,116)(96,126,160,117)(97,127,141,118)(98,128,142,119)(99,129,143,120)(100,130,144,101), (1,113,40,122)(2,104,21,133)(3,115,22,124)(4,106,23,135)(5,117,24,126)(6,108,25,137)(7,119,26,128)(8,110,27,139)(9,101,28,130)(10,112,29,121)(11,103,30,132)(12,114,31,123)(13,105,32,134)(14,116,33,125)(15,107,34,136)(16,118,35,127)(17,109,36,138)(18,120,37,129)(19,111,38,140)(20,102,39,131)(41,93,67,157)(42,84,68,148)(43,95,69,159)(44,86,70,150)(45,97,71,141)(46,88,72,152)(47,99,73,143)(48,90,74,154)(49,81,75,145)(50,92,76,156)(51,83,77,147)(52,94,78,158)(53,85,79,149)(54,96,80,160)(55,87,61,151)(56,98,62,142)(57,89,63,153)(58,100,64,144)(59,91,65,155)(60,82,66,146)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,65)(62,64)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(81,87)(82,86)(83,85)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(101,109)(102,108)(103,107)(104,106)(110,120)(111,119)(112,118)(113,117)(114,116)(121,127)(122,126)(123,125)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159), (1,76,40,50)(2,77,21,51)(3,78,22,52)(4,79,23,53)(5,80,24,54)(6,61,25,55)(7,62,26,56)(8,63,27,57)(9,64,28,58)(10,65,29,59)(11,66,30,60)(12,67,31,41)(13,68,32,42)(14,69,33,43)(15,70,34,44)(16,71,35,45)(17,72,36,46)(18,73,37,47)(19,74,38,48)(20,75,39,49)(81,131,145,102)(82,132,146,103)(83,133,147,104)(84,134,148,105)(85,135,149,106)(86,136,150,107)(87,137,151,108)(88,138,152,109)(89,139,153,110)(90,140,154,111)(91,121,155,112)(92,122,156,113)(93,123,157,114)(94,124,158,115)(95,125,159,116)(96,126,160,117)(97,127,141,118)(98,128,142,119)(99,129,143,120)(100,130,144,101), (1,113,40,122)(2,104,21,133)(3,115,22,124)(4,106,23,135)(5,117,24,126)(6,108,25,137)(7,119,26,128)(8,110,27,139)(9,101,28,130)(10,112,29,121)(11,103,30,132)(12,114,31,123)(13,105,32,134)(14,116,33,125)(15,107,34,136)(16,118,35,127)(17,109,36,138)(18,120,37,129)(19,111,38,140)(20,102,39,131)(41,93,67,157)(42,84,68,148)(43,95,69,159)(44,86,70,150)(45,97,71,141)(46,88,72,152)(47,99,73,143)(48,90,74,154)(49,81,75,145)(50,92,76,156)(51,83,77,147)(52,94,78,158)(53,85,79,149)(54,96,80,160)(55,87,61,151)(56,98,62,142)(57,89,63,153)(58,100,64,144)(59,91,65,155)(60,82,66,146) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(34,40),(35,39),(36,38),(41,53),(42,52),(43,51),(44,50),(45,49),(46,48),(54,60),(55,59),(56,58),(61,65),(62,64),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(81,87),(82,86),(83,85),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95),(101,109),(102,108),(103,107),(104,106),(110,120),(111,119),(112,118),(113,117),(114,116),(121,127),(122,126),(123,125),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135),(141,155),(142,154),(143,153),(144,152),(145,151),(146,150),(147,149),(156,160),(157,159)], [(1,76,40,50),(2,77,21,51),(3,78,22,52),(4,79,23,53),(5,80,24,54),(6,61,25,55),(7,62,26,56),(8,63,27,57),(9,64,28,58),(10,65,29,59),(11,66,30,60),(12,67,31,41),(13,68,32,42),(14,69,33,43),(15,70,34,44),(16,71,35,45),(17,72,36,46),(18,73,37,47),(19,74,38,48),(20,75,39,49),(81,131,145,102),(82,132,146,103),(83,133,147,104),(84,134,148,105),(85,135,149,106),(86,136,150,107),(87,137,151,108),(88,138,152,109),(89,139,153,110),(90,140,154,111),(91,121,155,112),(92,122,156,113),(93,123,157,114),(94,124,158,115),(95,125,159,116),(96,126,160,117),(97,127,141,118),(98,128,142,119),(99,129,143,120),(100,130,144,101)], [(1,113,40,122),(2,104,21,133),(3,115,22,124),(4,106,23,135),(5,117,24,126),(6,108,25,137),(7,119,26,128),(8,110,27,139),(9,101,28,130),(10,112,29,121),(11,103,30,132),(12,114,31,123),(13,105,32,134),(14,116,33,125),(15,107,34,136),(16,118,35,127),(17,109,36,138),(18,120,37,129),(19,111,38,140),(20,102,39,131),(41,93,67,157),(42,84,68,148),(43,95,69,159),(44,86,70,150),(45,97,71,141),(46,88,72,152),(47,99,73,143),(48,90,74,154),(49,81,75,145),(50,92,76,156),(51,83,77,147),(52,94,78,158),(53,85,79,149),(54,96,80,160),(55,87,61,151),(56,98,62,142),(57,89,63,153),(58,100,64,144),(59,91,65,155),(60,82,66,146)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D5 | D10 | D10 | D10 | 2- 1+4 | D4×D5 | Q8×D5 | Q8.10D10 |
| kernel | D20⋊8Q8 | C4×Dic10 | C4×D20 | C20⋊Q8 | D20⋊8C4 | D10⋊Q8 | D10⋊3Q8 | C5×C4⋊Q8 | C2×Q8×D5 | Dic10 | D20 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 8 | 4 | 1 | 4 | 4 | 4 |
Matrix representation of D20⋊8Q8 ►in GL6(𝔽41)
| 8 | 4 | 0 | 0 | 0 | 0 |
| 35 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 5 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 37 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 33 | 37 | 0 | 0 | 0 | 0 |
| 26 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 1 |
| 0 | 0 | 0 | 0 | 1 | 11 |
G:=sub<GL(6,GF(41))| [8,35,0,0,0,0,4,33,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,37,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[33,26,0,0,0,0,37,8,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,1,0,0,0,0,1,11] >;
D20⋊8Q8 in GAP, Magma, Sage, TeX
D_{20}\rtimes_8Q_8 % in TeX
G:=Group("D20:8Q8"); // GroupNames label
G:=SmallGroup(320,1399);
// by ID
G=gap.SmallGroup(320,1399);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,1571,297,136,12550]);
// 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^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations