metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊5M4(2), C42.29D10, C4⋊C8⋊12D5, C5⋊2C8⋊27D4, C5⋊8(C8⋊9D4), (C4×D20).8C2, C10.77(C4×D4), C4.206(D4×D5), (C2×D20).20C4, C20.365(C2×D4), (C2×C8).217D10, C4⋊Dic5.24C4, D10⋊1C8⋊23C2, C10.36(C8○D4), (C4×C20).62C22, C42.D5⋊3C2, C2.17(D5×M4(2)), D10⋊C4.23C4, C20.335(C4○D4), C2.7(D20⋊8C4), (C2×C40).212C22, (C2×C20).833C23, C4.55(Q8⋊2D5), C10.62(C2×M4(2)), C2.14(D20.3C4), (D5×C2×C8)⋊23C2, (C5×C4⋊C8)⋊17C2, (C2×C4).35(C4×D5), (C2×C8⋊D5)⋊21C2, C22.111(C2×C4×D5), (C2×C20).216(C2×C4), (C2×C4×D5).349C22, (C2×Dic5).24(C2×C4), (C22×D5).75(C2×C4), (C2×C4).775(C22×D5), (C2×C10).189(C22×C4), (C2×C5⋊2C8).199C22, SmallGroup(320,463)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊5M4(2)
G = < a,b,c,d | a10=b2=c8=d2=1, bab=dad=a-1, ac=ca, cbc-1=a5b, dbd=a3b, dcd=c5 >
Subgroups: 446 in 124 conjugacy classes, 51 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊2C8, C5⋊2C8, C40, C4×D5, D20, C2×Dic5, C2×C20, C22×D5, C8⋊9D4, C8×D5, C8⋊D5, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C42.D5, D10⋊1C8, C5×C4⋊C8, C4×D20, D5×C2×C8, C2×C8⋊D5, D10⋊5M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, M4(2), C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×M4(2), C8○D4, C4×D5, C22×D5, C8⋊9D4, C2×C4×D5, D4×D5, Q8⋊2D5, D20⋊8C4, D20.3C4, D5×M4(2), D10⋊5M4(2)
►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)
(1 117)(2 116)(3 115)(4 114)(5 113)(6 112)(7 111)(8 120)(9 119)(10 118)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 70)(21 126)(22 125)(23 124)(24 123)(25 122)(26 121)(27 130)(28 129)(29 128)(30 127)(31 94)(32 93)(33 92)(34 91)(35 100)(36 99)(37 98)(38 97)(39 96)(40 95)(41 108)(42 107)(43 106)(44 105)(45 104)(46 103)(47 102)(48 101)(49 110)(50 109)(51 159)(52 158)(53 157)(54 156)(55 155)(56 154)(57 153)(58 152)(59 151)(60 160)(71 146)(72 145)(73 144)(74 143)(75 142)(76 141)(77 150)(78 149)(79 148)(80 147)(81 136)(82 135)(83 134)(84 133)(85 132)(86 131)(87 140)(88 139)(89 138)(90 137)
(1 145 45 11 23 138 38 158)(2 146 46 12 24 139 39 159)(3 147 47 13 25 140 40 160)(4 148 48 14 26 131 31 151)(5 149 49 15 27 132 32 152)(6 150 50 16 28 133 33 153)(7 141 41 17 29 134 34 154)(8 142 42 18 30 135 35 155)(9 143 43 19 21 136 36 156)(10 144 44 20 22 137 37 157)(51 111 71 108 68 128 88 91)(52 112 72 109 69 129 89 92)(53 113 73 110 70 130 90 93)(54 114 74 101 61 121 81 94)(55 115 75 102 62 122 82 95)(56 116 76 103 63 123 83 96)(57 117 77 104 64 124 84 97)(58 118 78 105 65 125 85 98)(59 119 79 106 66 126 86 99)(60 120 80 107 67 127 87 100)
(1 23)(2 22)(3 21)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(12 20)(13 19)(14 18)(15 17)(31 42)(32 41)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(51 60)(52 59)(53 58)(54 57)(55 56)(61 64)(62 63)(65 70)(66 69)(67 68)(71 80)(72 79)(73 78)(74 77)(75 76)(81 84)(82 83)(85 90)(86 89)(87 88)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 110)(99 109)(100 108)(111 127)(112 126)(113 125)(114 124)(115 123)(116 122)(117 121)(118 130)(119 129)(120 128)(131 135)(132 134)(136 140)(137 139)(141 149)(142 148)(143 147)(144 146)(151 155)(152 154)(156 160)(157 159)
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), (1,117)(2,116)(3,115)(4,114)(5,113)(6,112)(7,111)(8,120)(9,119)(10,118)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,70)(21,126)(22,125)(23,124)(24,123)(25,122)(26,121)(27,130)(28,129)(29,128)(30,127)(31,94)(32,93)(33,92)(34,91)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,110)(50,109)(51,159)(52,158)(53,157)(54,156)(55,155)(56,154)(57,153)(58,152)(59,151)(60,160)(71,146)(72,145)(73,144)(74,143)(75,142)(76,141)(77,150)(78,149)(79,148)(80,147)(81,136)(82,135)(83,134)(84,133)(85,132)(86,131)(87,140)(88,139)(89,138)(90,137), (1,145,45,11,23,138,38,158)(2,146,46,12,24,139,39,159)(3,147,47,13,25,140,40,160)(4,148,48,14,26,131,31,151)(5,149,49,15,27,132,32,152)(6,150,50,16,28,133,33,153)(7,141,41,17,29,134,34,154)(8,142,42,18,30,135,35,155)(9,143,43,19,21,136,36,156)(10,144,44,20,22,137,37,157)(51,111,71,108,68,128,88,91)(52,112,72,109,69,129,89,92)(53,113,73,110,70,130,90,93)(54,114,74,101,61,121,81,94)(55,115,75,102,62,122,82,95)(56,116,76,103,63,123,83,96)(57,117,77,104,64,124,84,97)(58,118,78,105,65,125,85,98)(59,119,79,106,66,126,86,99)(60,120,80,107,67,127,87,100), (1,23)(2,22)(3,21)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(12,20)(13,19)(14,18)(15,17)(31,42)(32,41)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(51,60)(52,59)(53,58)(54,57)(55,56)(61,64)(62,63)(65,70)(66,69)(67,68)(71,80)(72,79)(73,78)(74,77)(75,76)(81,84)(82,83)(85,90)(86,89)(87,88)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,110)(99,109)(100,108)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,130)(119,129)(120,128)(131,135)(132,134)(136,140)(137,139)(141,149)(142,148)(143,147)(144,146)(151,155)(152,154)(156,160)(157,159)>;
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), (1,117)(2,116)(3,115)(4,114)(5,113)(6,112)(7,111)(8,120)(9,119)(10,118)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,70)(21,126)(22,125)(23,124)(24,123)(25,122)(26,121)(27,130)(28,129)(29,128)(30,127)(31,94)(32,93)(33,92)(34,91)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,110)(50,109)(51,159)(52,158)(53,157)(54,156)(55,155)(56,154)(57,153)(58,152)(59,151)(60,160)(71,146)(72,145)(73,144)(74,143)(75,142)(76,141)(77,150)(78,149)(79,148)(80,147)(81,136)(82,135)(83,134)(84,133)(85,132)(86,131)(87,140)(88,139)(89,138)(90,137), (1,145,45,11,23,138,38,158)(2,146,46,12,24,139,39,159)(3,147,47,13,25,140,40,160)(4,148,48,14,26,131,31,151)(5,149,49,15,27,132,32,152)(6,150,50,16,28,133,33,153)(7,141,41,17,29,134,34,154)(8,142,42,18,30,135,35,155)(9,143,43,19,21,136,36,156)(10,144,44,20,22,137,37,157)(51,111,71,108,68,128,88,91)(52,112,72,109,69,129,89,92)(53,113,73,110,70,130,90,93)(54,114,74,101,61,121,81,94)(55,115,75,102,62,122,82,95)(56,116,76,103,63,123,83,96)(57,117,77,104,64,124,84,97)(58,118,78,105,65,125,85,98)(59,119,79,106,66,126,86,99)(60,120,80,107,67,127,87,100), (1,23)(2,22)(3,21)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(12,20)(13,19)(14,18)(15,17)(31,42)(32,41)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(51,60)(52,59)(53,58)(54,57)(55,56)(61,64)(62,63)(65,70)(66,69)(67,68)(71,80)(72,79)(73,78)(74,77)(75,76)(81,84)(82,83)(85,90)(86,89)(87,88)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,110)(99,109)(100,108)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,130)(119,129)(120,128)(131,135)(132,134)(136,140)(137,139)(141,149)(142,148)(143,147)(144,146)(151,155)(152,154)(156,160)(157,159) );
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)], [(1,117),(2,116),(3,115),(4,114),(5,113),(6,112),(7,111),(8,120),(9,119),(10,118),(11,69),(12,68),(13,67),(14,66),(15,65),(16,64),(17,63),(18,62),(19,61),(20,70),(21,126),(22,125),(23,124),(24,123),(25,122),(26,121),(27,130),(28,129),(29,128),(30,127),(31,94),(32,93),(33,92),(34,91),(35,100),(36,99),(37,98),(38,97),(39,96),(40,95),(41,108),(42,107),(43,106),(44,105),(45,104),(46,103),(47,102),(48,101),(49,110),(50,109),(51,159),(52,158),(53,157),(54,156),(55,155),(56,154),(57,153),(58,152),(59,151),(60,160),(71,146),(72,145),(73,144),(74,143),(75,142),(76,141),(77,150),(78,149),(79,148),(80,147),(81,136),(82,135),(83,134),(84,133),(85,132),(86,131),(87,140),(88,139),(89,138),(90,137)], [(1,145,45,11,23,138,38,158),(2,146,46,12,24,139,39,159),(3,147,47,13,25,140,40,160),(4,148,48,14,26,131,31,151),(5,149,49,15,27,132,32,152),(6,150,50,16,28,133,33,153),(7,141,41,17,29,134,34,154),(8,142,42,18,30,135,35,155),(9,143,43,19,21,136,36,156),(10,144,44,20,22,137,37,157),(51,111,71,108,68,128,88,91),(52,112,72,109,69,129,89,92),(53,113,73,110,70,130,90,93),(54,114,74,101,61,121,81,94),(55,115,75,102,62,122,82,95),(56,116,76,103,63,123,83,96),(57,117,77,104,64,124,84,97),(58,118,78,105,65,125,85,98),(59,119,79,106,66,126,86,99),(60,120,80,107,67,127,87,100)], [(1,23),(2,22),(3,21),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(12,20),(13,19),(14,18),(15,17),(31,42),(32,41),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(51,60),(52,59),(53,58),(54,57),(55,56),(61,64),(62,63),(65,70),(66,69),(67,68),(71,80),(72,79),(73,78),(74,77),(75,76),(81,84),(82,83),(85,90),(86,89),(87,88),(91,107),(92,106),(93,105),(94,104),(95,103),(96,102),(97,101),(98,110),(99,109),(100,108),(111,127),(112,126),(113,125),(114,124),(115,123),(116,122),(117,121),(118,130),(119,129),(120,128),(131,135),(132,134),(136,140),(137,139),(141,149),(142,148),(143,147),(144,146),(151,155),(152,154),(156,160),(157,159)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | C4○D4 | M4(2) | D10 | D10 | C8○D4 | C4×D5 | D20.3C4 | D4×D5 | Q8⋊2D5 | D5×M4(2) |
| kernel | D10⋊5M4(2) | C42.D5 | D10⋊1C8 | C5×C4⋊C8 | C4×D20 | D5×C2×C8 | C2×C8⋊D5 | C4⋊Dic5 | D10⋊C4 | C2×D20 | C5⋊2C8 | C4⋊C8 | C20 | D10 | C42 | C2×C8 | C10 | C2×C4 | C2 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 16 | 2 | 2 | 4 |
Matrix representation of D10⋊5M4(2) ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 27 | 32 | 0 | 0 | 0 | 0 |
| 8 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 2 |
| 0 | 0 | 0 | 0 | 37 | 3 |
| 32 | 3 | 0 | 0 | 0 | 0 |
| 17 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 3 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 3 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[27,8,0,0,0,0,32,14,0,0,0,0,0,0,40,34,0,0,0,0,0,1,0,0,0,0,0,0,38,37,0,0,0,0,2,3],[32,17,0,0,0,0,3,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,40],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,34,40,0,0,0,0,7,7,0,0,0,0,0,0,1,3,0,0,0,0,0,40] >;
D10⋊5M4(2) in GAP, Magma, Sage, TeX
D_{10}\rtimes_5M_4(2) % in TeX
G:=Group("D10:5M4(2)"); // GroupNames label
G:=SmallGroup(320,463);
// by ID
G=gap.SmallGroup(320,463);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,701,120,219,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^5*b,d*b*d=a^3*b,d*c*d=c^5>;
// generators/relations