metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2)⋊2Dic5, (C2×C20).9Q8, C20.40(C4⋊C4), (C2×C4).128D20, (C2×C20).468D4, C4.Dic5⋊11C4, C4.3(C4⋊Dic5), (C2×C4).4Dic10, C23.43(C4×D5), (C5×M4(2))⋊11C4, (C2×C10).25C42, (C22×C4).61D10, (C2×M4(2)).5D5, C22.3(C4×Dic5), C5⋊4(C22.C42), (C10×M4(2)).9C2, C4.17(C23.D5), C4.34(D10⋊C4), (C22×Dic5).3C4, C20.133(C22⋊C4), C4.24(C10.D4), C10.12(C4.D4), C2.2(C20.46D4), C2.2(C4.12D20), C10.10(C4.10D4), (C22×C20).123C22, C22.5(C10.D4), C22.41(D10⋊C4), C10.32(C2.C42), C2.13(C10.10C42), (C2×C4).20(C4×D5), (C2×C10).34(C4⋊C4), (C2×C20).234(C2×C4), (C2×C4).21(C5⋊D4), (C2×C4⋊Dic5).29C2, (C2×C4).14(C2×Dic5), (C22×C10).99(C2×C4), (C2×C4.Dic5).10C2, (C2×C10).117(C22⋊C4), SmallGroup(320,112)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)⋊Dic5
G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=a5, ac=ca, dad-1=ab, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 310 in 98 conjugacy classes, 51 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×C4⋊C4, C2×M4(2), C2×M4(2), C5⋊2C8, C40, C2×Dic5, C2×C20, C22×C10, C22.C42, C2×C5⋊2C8, C4.Dic5, C4.Dic5, C4⋊Dic5, C2×C40, C5×M4(2), C5×M4(2), C22×Dic5, C22×C20, C2×C4.Dic5, C2×C4⋊Dic5, C10×M4(2), M4(2)⋊Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, C4.D4, C4.10D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C22.C42, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C20.46D4, C4.12D20, C10.10C42, M4(2)⋊Dic5
►On 160 points
Generators in S160
(1 71 36 63 28 90 48 51)(2 72 37 64 29 81 49 52)(3 73 38 65 30 82 50 53)(4 74 39 66 21 83 41 54)(5 75 40 67 22 84 42 55)(6 76 31 68 23 85 43 56)(7 77 32 69 24 86 44 57)(8 78 33 70 25 87 45 58)(9 79 34 61 26 88 46 59)(10 80 35 62 27 89 47 60)(11 122 142 102 159 119 139 99)(12 123 143 103 160 120 140 100)(13 124 144 104 151 111 131 91)(14 125 145 105 152 112 132 92)(15 126 146 106 153 113 133 93)(16 127 147 107 154 114 134 94)(17 128 148 108 155 115 135 95)(18 129 149 109 156 116 136 96)(19 130 150 110 157 117 137 97)(20 121 141 101 158 118 138 98)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 160)(18 151)(19 152)(20 153)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 68)(52 69)(53 70)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 81)(78 82)(79 83)(80 84)(91 96)(92 97)(93 98)(94 99)(95 100)(101 106)(102 107)(103 108)(104 109)(105 110)(111 116)(112 117)(113 118)(114 119)(115 120)(121 126)(122 127)(123 128)(124 129)(125 130)(131 149)(132 150)(133 141)(134 142)(135 143)(136 144)(137 145)(138 146)(139 147)(140 148)
(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 99 6 94)(2 98 7 93)(3 97 8 92)(4 96 9 91)(5 95 10 100)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 134 56 139)(52 133 57 138)(53 132 58 137)(54 131 59 136)(55 140 60 135)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)
G:=sub<Sym(160)| (1,71,36,63,28,90,48,51)(2,72,37,64,29,81,49,52)(3,73,38,65,30,82,50,53)(4,74,39,66,21,83,41,54)(5,75,40,67,22,84,42,55)(6,76,31,68,23,85,43,56)(7,77,32,69,24,86,44,57)(8,78,33,70,25,87,45,58)(9,79,34,61,26,88,46,59)(10,80,35,62,27,89,47,60)(11,122,142,102,159,119,139,99)(12,123,143,103,160,120,140,100)(13,124,144,104,151,111,131,91)(14,125,145,105,152,112,132,92)(15,126,146,106,153,113,133,93)(16,127,147,107,154,114,134,94)(17,128,148,108,155,115,135,95)(18,129,149,109,156,116,136,96)(19,130,150,110,157,117,137,97)(20,121,141,101,158,118,138,98), (1,6)(2,7)(3,8)(4,9)(5,10)(11,154)(12,155)(13,156)(14,157)(15,158)(16,159)(17,160)(18,151)(19,152)(20,153)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,68)(52,69)(53,70)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,116)(112,117)(113,118)(114,119)(115,120)(121,126)(122,127)(123,128)(124,129)(125,130)(131,149)(132,150)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146)(139,147)(140,148), (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), (1,99,6,94)(2,98,7,93)(3,97,8,92)(4,96,9,91)(5,95,10,100)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155)>;
G:=Group( (1,71,36,63,28,90,48,51)(2,72,37,64,29,81,49,52)(3,73,38,65,30,82,50,53)(4,74,39,66,21,83,41,54)(5,75,40,67,22,84,42,55)(6,76,31,68,23,85,43,56)(7,77,32,69,24,86,44,57)(8,78,33,70,25,87,45,58)(9,79,34,61,26,88,46,59)(10,80,35,62,27,89,47,60)(11,122,142,102,159,119,139,99)(12,123,143,103,160,120,140,100)(13,124,144,104,151,111,131,91)(14,125,145,105,152,112,132,92)(15,126,146,106,153,113,133,93)(16,127,147,107,154,114,134,94)(17,128,148,108,155,115,135,95)(18,129,149,109,156,116,136,96)(19,130,150,110,157,117,137,97)(20,121,141,101,158,118,138,98), (1,6)(2,7)(3,8)(4,9)(5,10)(11,154)(12,155)(13,156)(14,157)(15,158)(16,159)(17,160)(18,151)(19,152)(20,153)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,68)(52,69)(53,70)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,116)(112,117)(113,118)(114,119)(115,120)(121,126)(122,127)(123,128)(124,129)(125,130)(131,149)(132,150)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146)(139,147)(140,148), (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), (1,99,6,94)(2,98,7,93)(3,97,8,92)(4,96,9,91)(5,95,10,100)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155) );
G=PermutationGroup([[(1,71,36,63,28,90,48,51),(2,72,37,64,29,81,49,52),(3,73,38,65,30,82,50,53),(4,74,39,66,21,83,41,54),(5,75,40,67,22,84,42,55),(6,76,31,68,23,85,43,56),(7,77,32,69,24,86,44,57),(8,78,33,70,25,87,45,58),(9,79,34,61,26,88,46,59),(10,80,35,62,27,89,47,60),(11,122,142,102,159,119,139,99),(12,123,143,103,160,120,140,100),(13,124,144,104,151,111,131,91),(14,125,145,105,152,112,132,92),(15,126,146,106,153,113,133,93),(16,127,147,107,154,114,134,94),(17,128,148,108,155,115,135,95),(18,129,149,109,156,116,136,96),(19,130,150,110,157,117,137,97),(20,121,141,101,158,118,138,98)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,154),(12,155),(13,156),(14,157),(15,158),(16,159),(17,160),(18,151),(19,152),(20,153),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,46),(42,47),(43,48),(44,49),(45,50),(51,68),(52,69),(53,70),(54,61),(55,62),(56,63),(57,64),(58,65),(59,66),(60,67),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,81),(78,82),(79,83),(80,84),(91,96),(92,97),(93,98),(94,99),(95,100),(101,106),(102,107),(103,108),(104,109),(105,110),(111,116),(112,117),(113,118),(114,119),(115,120),(121,126),(122,127),(123,128),(124,129),(125,130),(131,149),(132,150),(133,141),(134,142),(135,143),(136,144),(137,145),(138,146),(139,147),(140,148)], [(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)], [(1,99,6,94),(2,98,7,93),(3,97,8,92),(4,96,9,91),(5,95,10,100),(11,90,16,85),(12,89,17,84),(13,88,18,83),(14,87,19,82),(15,86,20,81),(21,109,26,104),(22,108,27,103),(23,107,28,102),(24,106,29,101),(25,105,30,110),(31,114,36,119),(32,113,37,118),(33,112,38,117),(34,111,39,116),(35,120,40,115),(41,129,46,124),(42,128,47,123),(43,127,48,122),(44,126,49,121),(45,125,50,130),(51,134,56,139),(52,133,57,138),(53,132,58,137),(54,131,59,136),(55,140,60,135),(61,149,66,144),(62,148,67,143),(63,147,68,142),(64,146,69,141),(65,145,70,150),(71,154,76,159),(72,153,77,158),(73,152,78,157),(74,151,79,156),(75,160,80,155)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | - | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D5 | Dic5 | D10 | Dic10 | C4×D5 | D20 | C5⋊D4 | C4×D5 | C4.D4 | C4.10D4 | C20.46D4 | C4.12D20 |
| kernel | M4(2)⋊Dic5 | C2×C4.Dic5 | C2×C4⋊Dic5 | C10×M4(2) | C4.Dic5 | C5×M4(2) | C22×Dic5 | C2×C20 | C2×C20 | C2×M4(2) | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 3 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 4 | 1 | 1 | 4 | 4 |
Matrix representation of M4(2)⋊Dic5 ►in GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 11 | 32 | 0 | 0 |
| 0 | 0 | 9 | 30 | 0 | 0 |
| 1 | 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 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 25 | 0 | 0 | 0 | 0 | 0 |
| 2 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 6 | 35 | 0 | 0 | 0 | 0 |
| 13 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 21 | 0 | 0 |
| 0 | 0 | 35 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 21 |
| 0 | 0 | 0 | 0 | 35 | 39 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,11,9,0,0,0,0,32,30,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[25,2,0,0,0,0,0,23,0,0,0,0,0,0,34,40,0,0,0,0,1,0,0,0,0,0,0,0,34,40,0,0,0,0,1,0],[6,13,0,0,0,0,35,35,0,0,0,0,0,0,2,35,0,0,0,0,21,39,0,0,0,0,0,0,2,35,0,0,0,0,21,39] >;
M4(2)⋊Dic5 in GAP, Magma, Sage, TeX
M_4(2)\rtimes {\rm Dic}_5 % in TeX
G:=Group("M4(2):Dic5"); // GroupNames label
G:=SmallGroup(320,112);
// by ID
G=gap.SmallGroup(320,112);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,1123,136,851,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=c^5,b*a*b=a^5,a*c=c*a,d*a*d^-1=a*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations