metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊2M4(2), C5⋊C8⋊6D4, C4⋊C4.5F5, C5⋊3(C8⋊9D4), C10.9(C4×D4), C2.11(D4×F5), (C2×D20).8C4, D10⋊C8⋊9C2, C2.4(Q8.F5), Dic5⋊C8⋊2C2, D10⋊C4.9C4, C10.20(C8○D4), Dic5.70(C2×D4), D20⋊8C4.17C2, C10.C42⋊3C2, C10.13(C2×M4(2)), Dic5.55(C4○D4), C22.76(C22×F5), C2.13(D5⋊M4(2)), (C4×Dic5).68C22, (C2×Dic5).331C23, (C5×C4⋊C4).8C4, (C2×D5⋊C8)⋊11C2, (C2×C4.F5)⋊11C2, (C2×C4).25(C2×F5), (C2×C20).22(C2×C4), (C2×C5⋊C8).28C22, (C2×C4×D5).290C22, (C2×C10).42(C22×C4), (C2×Dic5).57(C2×C4), (C22×D5).48(C2×C4), SmallGroup(320,1042)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 474 in 124 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C2 [×3], C4 [×6], C22, C22 [×7], C5, C8 [×5], C2×C4 [×3], C2×C4 [×6], D4 [×2], C23 [×2], D5 [×3], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×6], M4(2) [×2], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5, C20 [×3], D10 [×2], D10 [×5], C2×C10, C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊C8 [×2], C5⋊C8 [×3], C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C8⋊9D4, C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4, D5⋊C8 [×2], C4.F5 [×2], C2×C5⋊C8 [×4], C2×C4×D5 [×2], C2×D20, C10.C42, D10⋊C8 [×2], Dic5⋊C8, D20⋊8C4, C2×D5⋊C8, C2×C4.F5, D10⋊2M4(2)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5 [×3], C8⋊9D4, C22×F5, D5⋊M4(2), D4×F5, Q8.F5, D10⋊2M4(2)
Generators and relations
G = < a,b,c,d | a10=b2=c8=d2=1, bab=dad=a-1, cac-1=a7, cbc-1=ab, dbd=a3b, dcd=c5 >
►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 79)(2 78)(3 77)(4 76)(5 75)(6 74)(7 73)(8 72)(9 71)(10 80)(11 107)(12 106)(13 105)(14 104)(15 103)(16 102)(17 101)(18 110)(19 109)(20 108)(21 85)(22 84)(23 83)(24 82)(25 81)(26 90)(27 89)(28 88)(29 87)(30 86)(31 69)(32 68)(33 67)(34 66)(35 65)(36 64)(37 63)(38 62)(39 61)(40 70)(41 52)(42 51)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(91 158)(92 157)(93 156)(94 155)(95 154)(96 153)(97 152)(98 151)(99 160)(100 159)(111 143)(112 142)(113 141)(114 150)(115 149)(116 148)(117 147)(118 146)(119 145)(120 144)(121 138)(122 137)(123 136)(124 135)(125 134)(126 133)(127 132)(128 131)(129 140)(130 139)
(1 120 40 105 21 130 44 100)(2 113 39 102 22 123 43 97)(3 116 38 109 23 126 42 94)(4 119 37 106 24 129 41 91)(5 112 36 103 25 122 50 98)(6 115 35 110 26 125 49 95)(7 118 34 107 27 128 48 92)(8 111 33 104 28 121 47 99)(9 114 32 101 29 124 46 96)(10 117 31 108 30 127 45 93)(11 82 133 54 157 76 148 65)(12 85 132 51 158 79 147 62)(13 88 131 58 159 72 146 69)(14 81 140 55 160 75 145 66)(15 84 139 52 151 78 144 63)(16 87 138 59 152 71 143 70)(17 90 137 56 153 74 142 67)(18 83 136 53 154 77 141 64)(19 86 135 60 155 80 150 61)(20 89 134 57 156 73 149 68)
(2 10)(3 9)(4 8)(5 7)(11 158)(12 157)(13 156)(14 155)(15 154)(16 153)(17 152)(18 151)(19 160)(20 159)(22 30)(23 29)(24 28)(25 27)(31 39)(32 38)(33 37)(34 36)(41 47)(42 46)(43 45)(48 50)(51 54)(52 53)(55 60)(56 59)(57 58)(61 66)(62 65)(63 64)(67 70)(68 69)(71 74)(72 73)(75 80)(76 79)(77 78)(81 86)(82 85)(83 84)(87 90)(88 89)(91 104)(92 103)(93 102)(94 101)(95 110)(96 109)(97 108)(98 107)(99 106)(100 105)(111 129)(112 128)(113 127)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)(120 130)(131 149)(132 148)(133 147)(134 146)(135 145)(136 144)(137 143)(138 142)(139 141)(140 150)
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), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,72)(9,71)(10,80)(11,107)(12,106)(13,105)(14,104)(15,103)(16,102)(17,101)(18,110)(19,109)(20,108)(21,85)(22,84)(23,83)(24,82)(25,81)(26,90)(27,89)(28,88)(29,87)(30,86)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64)(37,63)(38,62)(39,61)(40,70)(41,52)(42,51)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(91,158)(92,157)(93,156)(94,155)(95,154)(96,153)(97,152)(98,151)(99,160)(100,159)(111,143)(112,142)(113,141)(114,150)(115,149)(116,148)(117,147)(118,146)(119,145)(120,144)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,140)(130,139), (1,120,40,105,21,130,44,100)(2,113,39,102,22,123,43,97)(3,116,38,109,23,126,42,94)(4,119,37,106,24,129,41,91)(5,112,36,103,25,122,50,98)(6,115,35,110,26,125,49,95)(7,118,34,107,27,128,48,92)(8,111,33,104,28,121,47,99)(9,114,32,101,29,124,46,96)(10,117,31,108,30,127,45,93)(11,82,133,54,157,76,148,65)(12,85,132,51,158,79,147,62)(13,88,131,58,159,72,146,69)(14,81,140,55,160,75,145,66)(15,84,139,52,151,78,144,63)(16,87,138,59,152,71,143,70)(17,90,137,56,153,74,142,67)(18,83,136,53,154,77,141,64)(19,86,135,60,155,80,150,61)(20,89,134,57,156,73,149,68), (2,10)(3,9)(4,8)(5,7)(11,158)(12,157)(13,156)(14,155)(15,154)(16,153)(17,152)(18,151)(19,160)(20,159)(22,30)(23,29)(24,28)(25,27)(31,39)(32,38)(33,37)(34,36)(41,47)(42,46)(43,45)(48,50)(51,54)(52,53)(55,60)(56,59)(57,58)(61,66)(62,65)(63,64)(67,70)(68,69)(71,74)(72,73)(75,80)(76,79)(77,78)(81,86)(82,85)(83,84)(87,90)(88,89)(91,104)(92,103)(93,102)(94,101)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(111,129)(112,128)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(120,130)(131,149)(132,148)(133,147)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141)(140,150)>;
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), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,72)(9,71)(10,80)(11,107)(12,106)(13,105)(14,104)(15,103)(16,102)(17,101)(18,110)(19,109)(20,108)(21,85)(22,84)(23,83)(24,82)(25,81)(26,90)(27,89)(28,88)(29,87)(30,86)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64)(37,63)(38,62)(39,61)(40,70)(41,52)(42,51)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(91,158)(92,157)(93,156)(94,155)(95,154)(96,153)(97,152)(98,151)(99,160)(100,159)(111,143)(112,142)(113,141)(114,150)(115,149)(116,148)(117,147)(118,146)(119,145)(120,144)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,140)(130,139), (1,120,40,105,21,130,44,100)(2,113,39,102,22,123,43,97)(3,116,38,109,23,126,42,94)(4,119,37,106,24,129,41,91)(5,112,36,103,25,122,50,98)(6,115,35,110,26,125,49,95)(7,118,34,107,27,128,48,92)(8,111,33,104,28,121,47,99)(9,114,32,101,29,124,46,96)(10,117,31,108,30,127,45,93)(11,82,133,54,157,76,148,65)(12,85,132,51,158,79,147,62)(13,88,131,58,159,72,146,69)(14,81,140,55,160,75,145,66)(15,84,139,52,151,78,144,63)(16,87,138,59,152,71,143,70)(17,90,137,56,153,74,142,67)(18,83,136,53,154,77,141,64)(19,86,135,60,155,80,150,61)(20,89,134,57,156,73,149,68), (2,10)(3,9)(4,8)(5,7)(11,158)(12,157)(13,156)(14,155)(15,154)(16,153)(17,152)(18,151)(19,160)(20,159)(22,30)(23,29)(24,28)(25,27)(31,39)(32,38)(33,37)(34,36)(41,47)(42,46)(43,45)(48,50)(51,54)(52,53)(55,60)(56,59)(57,58)(61,66)(62,65)(63,64)(67,70)(68,69)(71,74)(72,73)(75,80)(76,79)(77,78)(81,86)(82,85)(83,84)(87,90)(88,89)(91,104)(92,103)(93,102)(94,101)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(111,129)(112,128)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(120,130)(131,149)(132,148)(133,147)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141)(140,150) );
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)], [(1,79),(2,78),(3,77),(4,76),(5,75),(6,74),(7,73),(8,72),(9,71),(10,80),(11,107),(12,106),(13,105),(14,104),(15,103),(16,102),(17,101),(18,110),(19,109),(20,108),(21,85),(22,84),(23,83),(24,82),(25,81),(26,90),(27,89),(28,88),(29,87),(30,86),(31,69),(32,68),(33,67),(34,66),(35,65),(36,64),(37,63),(38,62),(39,61),(40,70),(41,52),(42,51),(43,60),(44,59),(45,58),(46,57),(47,56),(48,55),(49,54),(50,53),(91,158),(92,157),(93,156),(94,155),(95,154),(96,153),(97,152),(98,151),(99,160),(100,159),(111,143),(112,142),(113,141),(114,150),(115,149),(116,148),(117,147),(118,146),(119,145),(120,144),(121,138),(122,137),(123,136),(124,135),(125,134),(126,133),(127,132),(128,131),(129,140),(130,139)], [(1,120,40,105,21,130,44,100),(2,113,39,102,22,123,43,97),(3,116,38,109,23,126,42,94),(4,119,37,106,24,129,41,91),(5,112,36,103,25,122,50,98),(6,115,35,110,26,125,49,95),(7,118,34,107,27,128,48,92),(8,111,33,104,28,121,47,99),(9,114,32,101,29,124,46,96),(10,117,31,108,30,127,45,93),(11,82,133,54,157,76,148,65),(12,85,132,51,158,79,147,62),(13,88,131,58,159,72,146,69),(14,81,140,55,160,75,145,66),(15,84,139,52,151,78,144,63),(16,87,138,59,152,71,143,70),(17,90,137,56,153,74,142,67),(18,83,136,53,154,77,141,64),(19,86,135,60,155,80,150,61),(20,89,134,57,156,73,149,68)], [(2,10),(3,9),(4,8),(5,7),(11,158),(12,157),(13,156),(14,155),(15,154),(16,153),(17,152),(18,151),(19,160),(20,159),(22,30),(23,29),(24,28),(25,27),(31,39),(32,38),(33,37),(34,36),(41,47),(42,46),(43,45),(48,50),(51,54),(52,53),(55,60),(56,59),(57,58),(61,66),(62,65),(63,64),(67,70),(68,69),(71,74),(72,73),(75,80),(76,79),(77,78),(81,86),(82,85),(83,84),(87,90),(88,89),(91,104),(92,103),(93,102),(94,101),(95,110),(96,109),(97,108),(98,107),(99,106),(100,105),(111,129),(112,128),(113,127),(114,126),(115,125),(116,124),(117,123),(118,122),(119,121),(120,130),(131,149),(132,148),(133,147),(134,146),(135,145),(136,144),(137,143),(138,142),(139,141),(140,150)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 | 40 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 22 | 19 | 3 |
| 0 | 0 | 19 | 0 | 22 | 3 |
| 0 | 0 | 38 | 3 | 22 | 0 |
| 0 | 0 | 0 | 3 | 19 | 22 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 9 | 11 | 11 |
| 0 | 0 | 12 | 18 | 12 | 20 |
| 0 | 0 | 21 | 29 | 23 | 29 |
| 0 | 0 | 30 | 30 | 32 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40,0,0,1,0,0,0],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,38,19,38,0,0,0,22,0,3,3,0,0,19,22,22,19,0,0,3,3,0,22],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,21,30,0,0,9,18,29,30,0,0,11,12,23,32,0,0,11,20,29,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 20 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 8 | ··· | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | C4○D4 | M4(2) | C8○D4 | F5 | C2×F5 | D5⋊M4(2) | D4×F5 | Q8.F5 |
| kernel | D10⋊2M4(2) | C10.C42 | D10⋊C8 | Dic5⋊C8 | D20⋊8C4 | C2×D5⋊C8 | C2×C4.F5 | D10⋊C4 | C5×C4⋊C4 | C2×D20 | C5⋊C8 | Dic5 | D10 | C10 | C4⋊C4 | C2×C4 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 3 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
D_{10}\rtimes_2M_{4(2)} % in TeX
G:=Group("D10:2M4(2)"); // GroupNames label
G:=SmallGroup(320,1042);
// by ID
G=gap.SmallGroup(320,1042);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,219,184,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=a*b,d*b*d=a^3*b,d*c*d=c^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀