metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊22D4, C10.192- 1+4, C5⋊4(Q8⋊5D4), C4.114(D4×D5), C22⋊Q8⋊10D5, C4⋊D20⋊26C2, C4⋊C4.191D10, C20.237(C2×D4), D10⋊14(C4○D4), D10⋊D4⋊26C2, D20⋊8C4⋊27C2, D10⋊Q8⋊20C2, (C2×C20).56C23, (C2×Q8).128D10, C22⋊C4.17D10, Dic5.49(C2×D4), C10.79(C22×D4), Dic5⋊3Q8⋊26C2, C20.23D4⋊13C2, (C2×C10).177C24, (C22×C4).239D10, Dic5.5D4⋊25C2, (C2×D20).273C22, (Q8×C10).109C22, C22.198(C23×D5), C23.120(C22×D5), (C22×C10).205C23, (C22×C20).257C22, (C2×Dic5).246C23, (C4×Dic5).115C22, C10.D4.29C22, (C22×D5).209C23, C23.D5.118C22, D10⋊C4.128C22, C2.20(Q8.10D10), (C2×Dic10).303C22, (C2×Q8×D5)⋊8C2, C2.52(C2×D4×D5), (C4×C5⋊D4)⋊24C2, C2.50(D5×C4○D4), (C2×C4○D20)⋊25C2, (C5×C22⋊Q8)⋊13C2, C10.162(C2×C4○D4), (C2×C4×D5).106C22, (C5×C4⋊C4).160C22, (C2×C4).592(C22×D5), (C2×C5⋊D4).133C22, (C5×C22⋊C4).32C22, SmallGroup(320,1305)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊22D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=dad=a9, cbc-1=a10b, bd=db, dcd=c-1 >
Subgroups: 1078 in 290 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, Dic10, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C22×C10, Q8⋊5D4, C4×Dic5, C4×Dic5, C10.D4, C10.D4, D10⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C4○D20, Q8×D5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, Q8×C10, D10⋊D4, Dic5.5D4, Dic5⋊3Q8, D20⋊8C4, C4⋊D20, D10⋊Q8, C4×C5⋊D4, C20.23D4, C5×C22⋊Q8, C2×C4○D20, C2×Q8×D5, Dic10⋊22D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2- 1+4, C22×D5, Q8⋊5D4, D4×D5, C23×D5, C2×D4×D5, Q8.10D10, D5×C4○D4, Dic10⋊22D4
►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 49 11 59)(2 48 12 58)(3 47 13 57)(4 46 14 56)(5 45 15 55)(6 44 16 54)(7 43 17 53)(8 42 18 52)(9 41 19 51)(10 60 20 50)(21 139 31 129)(22 138 32 128)(23 137 33 127)(24 136 34 126)(25 135 35 125)(26 134 36 124)(27 133 37 123)(28 132 38 122)(29 131 39 121)(30 130 40 140)(61 101 71 111)(62 120 72 110)(63 119 73 109)(64 118 74 108)(65 117 75 107)(66 116 76 106)(67 115 77 105)(68 114 78 104)(69 113 79 103)(70 112 80 102)(81 147 91 157)(82 146 92 156)(83 145 93 155)(84 144 94 154)(85 143 95 153)(86 142 96 152)(87 141 97 151)(88 160 98 150)(89 159 99 149)(90 158 100 148)
(1 86 127 112)(2 95 128 101)(3 84 129 110)(4 93 130 119)(5 82 131 108)(6 91 132 117)(7 100 133 106)(8 89 134 115)(9 98 135 104)(10 87 136 113)(11 96 137 102)(12 85 138 111)(13 94 139 120)(14 83 140 109)(15 92 121 118)(16 81 122 107)(17 90 123 116)(18 99 124 105)(19 88 125 114)(20 97 126 103)(21 72 47 154)(22 61 48 143)(23 70 49 152)(24 79 50 141)(25 68 51 150)(26 77 52 159)(27 66 53 148)(28 75 54 157)(29 64 55 146)(30 73 56 155)(31 62 57 144)(32 71 58 153)(33 80 59 142)(34 69 60 151)(35 78 41 160)(36 67 42 149)(37 76 43 158)(38 65 44 147)(39 74 45 156)(40 63 46 145)
(1 70)(2 79)(3 68)(4 77)(5 66)(6 75)(7 64)(8 73)(9 62)(10 71)(11 80)(12 69)(13 78)(14 67)(15 76)(16 65)(17 74)(18 63)(19 72)(20 61)(21 88)(22 97)(23 86)(24 95)(25 84)(26 93)(27 82)(28 91)(29 100)(30 89)(31 98)(32 87)(33 96)(34 85)(35 94)(36 83)(37 92)(38 81)(39 90)(40 99)(41 120)(42 109)(43 118)(44 107)(45 116)(46 105)(47 114)(48 103)(49 112)(50 101)(51 110)(52 119)(53 108)(54 117)(55 106)(56 115)(57 104)(58 113)(59 102)(60 111)(121 158)(122 147)(123 156)(124 145)(125 154)(126 143)(127 152)(128 141)(129 150)(130 159)(131 148)(132 157)(133 146)(134 155)(135 144)(136 153)(137 142)(138 151)(139 160)(140 149)
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), (1,49,11,59)(2,48,12,58)(3,47,13,57)(4,46,14,56)(5,45,15,55)(6,44,16,54)(7,43,17,53)(8,42,18,52)(9,41,19,51)(10,60,20,50)(21,139,31,129)(22,138,32,128)(23,137,33,127)(24,136,34,126)(25,135,35,125)(26,134,36,124)(27,133,37,123)(28,132,38,122)(29,131,39,121)(30,130,40,140)(61,101,71,111)(62,120,72,110)(63,119,73,109)(64,118,74,108)(65,117,75,107)(66,116,76,106)(67,115,77,105)(68,114,78,104)(69,113,79,103)(70,112,80,102)(81,147,91,157)(82,146,92,156)(83,145,93,155)(84,144,94,154)(85,143,95,153)(86,142,96,152)(87,141,97,151)(88,160,98,150)(89,159,99,149)(90,158,100,148), (1,86,127,112)(2,95,128,101)(3,84,129,110)(4,93,130,119)(5,82,131,108)(6,91,132,117)(7,100,133,106)(8,89,134,115)(9,98,135,104)(10,87,136,113)(11,96,137,102)(12,85,138,111)(13,94,139,120)(14,83,140,109)(15,92,121,118)(16,81,122,107)(17,90,123,116)(18,99,124,105)(19,88,125,114)(20,97,126,103)(21,72,47,154)(22,61,48,143)(23,70,49,152)(24,79,50,141)(25,68,51,150)(26,77,52,159)(27,66,53,148)(28,75,54,157)(29,64,55,146)(30,73,56,155)(31,62,57,144)(32,71,58,153)(33,80,59,142)(34,69,60,151)(35,78,41,160)(36,67,42,149)(37,76,43,158)(38,65,44,147)(39,74,45,156)(40,63,46,145), (1,70)(2,79)(3,68)(4,77)(5,66)(6,75)(7,64)(8,73)(9,62)(10,71)(11,80)(12,69)(13,78)(14,67)(15,76)(16,65)(17,74)(18,63)(19,72)(20,61)(21,88)(22,97)(23,86)(24,95)(25,84)(26,93)(27,82)(28,91)(29,100)(30,89)(31,98)(32,87)(33,96)(34,85)(35,94)(36,83)(37,92)(38,81)(39,90)(40,99)(41,120)(42,109)(43,118)(44,107)(45,116)(46,105)(47,114)(48,103)(49,112)(50,101)(51,110)(52,119)(53,108)(54,117)(55,106)(56,115)(57,104)(58,113)(59,102)(60,111)(121,158)(122,147)(123,156)(124,145)(125,154)(126,143)(127,152)(128,141)(129,150)(130,159)(131,148)(132,157)(133,146)(134,155)(135,144)(136,153)(137,142)(138,151)(139,160)(140,149)>;
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), (1,49,11,59)(2,48,12,58)(3,47,13,57)(4,46,14,56)(5,45,15,55)(6,44,16,54)(7,43,17,53)(8,42,18,52)(9,41,19,51)(10,60,20,50)(21,139,31,129)(22,138,32,128)(23,137,33,127)(24,136,34,126)(25,135,35,125)(26,134,36,124)(27,133,37,123)(28,132,38,122)(29,131,39,121)(30,130,40,140)(61,101,71,111)(62,120,72,110)(63,119,73,109)(64,118,74,108)(65,117,75,107)(66,116,76,106)(67,115,77,105)(68,114,78,104)(69,113,79,103)(70,112,80,102)(81,147,91,157)(82,146,92,156)(83,145,93,155)(84,144,94,154)(85,143,95,153)(86,142,96,152)(87,141,97,151)(88,160,98,150)(89,159,99,149)(90,158,100,148), (1,86,127,112)(2,95,128,101)(3,84,129,110)(4,93,130,119)(5,82,131,108)(6,91,132,117)(7,100,133,106)(8,89,134,115)(9,98,135,104)(10,87,136,113)(11,96,137,102)(12,85,138,111)(13,94,139,120)(14,83,140,109)(15,92,121,118)(16,81,122,107)(17,90,123,116)(18,99,124,105)(19,88,125,114)(20,97,126,103)(21,72,47,154)(22,61,48,143)(23,70,49,152)(24,79,50,141)(25,68,51,150)(26,77,52,159)(27,66,53,148)(28,75,54,157)(29,64,55,146)(30,73,56,155)(31,62,57,144)(32,71,58,153)(33,80,59,142)(34,69,60,151)(35,78,41,160)(36,67,42,149)(37,76,43,158)(38,65,44,147)(39,74,45,156)(40,63,46,145), (1,70)(2,79)(3,68)(4,77)(5,66)(6,75)(7,64)(8,73)(9,62)(10,71)(11,80)(12,69)(13,78)(14,67)(15,76)(16,65)(17,74)(18,63)(19,72)(20,61)(21,88)(22,97)(23,86)(24,95)(25,84)(26,93)(27,82)(28,91)(29,100)(30,89)(31,98)(32,87)(33,96)(34,85)(35,94)(36,83)(37,92)(38,81)(39,90)(40,99)(41,120)(42,109)(43,118)(44,107)(45,116)(46,105)(47,114)(48,103)(49,112)(50,101)(51,110)(52,119)(53,108)(54,117)(55,106)(56,115)(57,104)(58,113)(59,102)(60,111)(121,158)(122,147)(123,156)(124,145)(125,154)(126,143)(127,152)(128,141)(129,150)(130,159)(131,148)(132,157)(133,146)(134,155)(135,144)(136,153)(137,142)(138,151)(139,160)(140,149) );
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)], [(1,49,11,59),(2,48,12,58),(3,47,13,57),(4,46,14,56),(5,45,15,55),(6,44,16,54),(7,43,17,53),(8,42,18,52),(9,41,19,51),(10,60,20,50),(21,139,31,129),(22,138,32,128),(23,137,33,127),(24,136,34,126),(25,135,35,125),(26,134,36,124),(27,133,37,123),(28,132,38,122),(29,131,39,121),(30,130,40,140),(61,101,71,111),(62,120,72,110),(63,119,73,109),(64,118,74,108),(65,117,75,107),(66,116,76,106),(67,115,77,105),(68,114,78,104),(69,113,79,103),(70,112,80,102),(81,147,91,157),(82,146,92,156),(83,145,93,155),(84,144,94,154),(85,143,95,153),(86,142,96,152),(87,141,97,151),(88,160,98,150),(89,159,99,149),(90,158,100,148)], [(1,86,127,112),(2,95,128,101),(3,84,129,110),(4,93,130,119),(5,82,131,108),(6,91,132,117),(7,100,133,106),(8,89,134,115),(9,98,135,104),(10,87,136,113),(11,96,137,102),(12,85,138,111),(13,94,139,120),(14,83,140,109),(15,92,121,118),(16,81,122,107),(17,90,123,116),(18,99,124,105),(19,88,125,114),(20,97,126,103),(21,72,47,154),(22,61,48,143),(23,70,49,152),(24,79,50,141),(25,68,51,150),(26,77,52,159),(27,66,53,148),(28,75,54,157),(29,64,55,146),(30,73,56,155),(31,62,57,144),(32,71,58,153),(33,80,59,142),(34,69,60,151),(35,78,41,160),(36,67,42,149),(37,76,43,158),(38,65,44,147),(39,74,45,156),(40,63,46,145)], [(1,70),(2,79),(3,68),(4,77),(5,66),(6,75),(7,64),(8,73),(9,62),(10,71),(11,80),(12,69),(13,78),(14,67),(15,76),(16,65),(17,74),(18,63),(19,72),(20,61),(21,88),(22,97),(23,86),(24,95),(25,84),(26,93),(27,82),(28,91),(29,100),(30,89),(31,98),(32,87),(33,96),(34,85),(35,94),(36,83),(37,92),(38,81),(39,90),(40,99),(41,120),(42,109),(43,118),(44,107),(45,116),(46,105),(47,114),(48,103),(49,112),(50,101),(51,110),(52,119),(53,108),(54,117),(55,106),(56,115),(57,104),(58,113),(59,102),(60,111),(121,158),(122,147),(123,156),(124,145),(125,154),(126,143),(127,152),(128,141),(129,150),(130,159),(131,148),(132,157),(133,146),(134,155),(135,144),(136,153),(137,142),(138,151),(139,160),(140,149)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2- 1+4 | D4×D5 | Q8.10D10 | D5×C4○D4 |
| kernel | Dic10⋊22D4 | D10⋊D4 | Dic5.5D4 | Dic5⋊3Q8 | D20⋊8C4 | C4⋊D20 | D10⋊Q8 | C4×C5⋊D4 | C20.23D4 | C5×C22⋊Q8 | C2×C4○D20 | C2×Q8×D5 | Dic10 | C22⋊Q8 | D10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 6 | 2 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of Dic10⋊22D4 ►in GL6(𝔽41)
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 23 |
| 0 | 0 | 0 | 0 | 32 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 18 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [0,9,0,0,0,0,9,0,0,0,0,0,0,0,6,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,35,1,0,0,0,0,6,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,6,40,0,0,0,0,35,35,0,0,0,0,0,0,1,32,0,0,0,0,23,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,1,0,0,0,0,6,6,0,0,0,0,0,0,40,0,0,0,0,0,18,1] >;
Dic10⋊22D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_{22}D_4 % in TeX
G:=Group("Dic10:22D4"); // GroupNames label
G:=SmallGroup(320,1305);
// by ID
G=gap.SmallGroup(320,1305);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,1571,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,c*a*c^-1=d*a*d=a^9,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations