direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4⋊Dic5, C22.15D20, C23.30D10, C22.5Dic10, C10⋊3(C4⋊C4), (C2×C20)⋊10C4, C20⋊10(C2×C4), (C2×C4)⋊3Dic5, C4⋊2(C2×Dic5), C2.2(C2×D20), C10.9(C2×Q8), (C2×C10).6Q8, C10.15(C2×D4), (C2×C10).20D4, (C2×C4).84D10, (C22×C4).6D5, (C22×C20).7C2, C2.3(C2×Dic10), (C2×C20).92C22, C10.36(C22×C4), (C2×C10).43C23, C2.4(C22×Dic5), (C22×Dic5).5C2, C22.14(C2×Dic5), C22.21(C22×D5), (C22×C10).35C22, (C2×Dic5).37C22, C5⋊4(C2×C4⋊C4), (C2×C10).54(C2×C4), SmallGroup(160,146)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4⋊Dic5
G = < a,b,c,d | a2=b4=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 216 in 92 conjugacy classes, 65 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C4⋊Dic5, C22×Dic5, C22×C20, C2×C4⋊Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, D10, C2×C4⋊C4, Dic10, D20, C2×Dic5, C22×D5, C4⋊Dic5, C2×Dic10, C2×D20, C22×Dic5, C2×C4⋊Dic5
►Regular action on 160 points
Generators in S160
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 61)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 111)(18 112)(19 113)(20 114)(21 59)(22 60)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 86)(32 87)(33 88)(34 89)(35 90)(36 81)(37 82)(38 83)(39 84)(40 85)(41 79)(42 80)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 141)(99 142)(100 143)(101 138)(102 139)(103 140)(104 131)(105 132)(106 133)(107 134)(108 135)(109 136)(110 137)(121 158)(122 159)(123 160)(124 151)(125 152)(126 153)(127 154)(128 155)(129 156)(130 157)
(1 47 27 40)(2 48 28 31)(3 49 29 32)(4 50 30 33)(5 41 21 34)(6 42 22 35)(7 43 23 36)(8 44 24 37)(9 45 25 38)(10 46 26 39)(11 140 160 148)(12 131 151 149)(13 132 152 150)(14 133 153 141)(15 134 154 142)(16 135 155 143)(17 136 156 144)(18 137 157 145)(19 138 158 146)(20 139 159 147)(51 81 68 71)(52 82 69 72)(53 83 70 73)(54 84 61 74)(55 85 62 75)(56 86 63 76)(57 87 64 77)(58 88 65 78)(59 89 66 79)(60 90 67 80)(91 111 109 129)(92 112 110 130)(93 113 101 121)(94 114 102 122)(95 115 103 123)(96 116 104 124)(97 117 105 125)(98 118 106 126)(99 119 107 127)(100 120 108 128)
(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 95 6 100)(2 94 7 99)(3 93 8 98)(4 92 9 97)(5 91 10 96)(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)| (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,61)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,111)(18,112)(19,113)(20,114)(21,59)(22,60)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,86)(32,87)(33,88)(34,89)(35,90)(36,81)(37,82)(38,83)(39,84)(40,85)(41,79)(42,80)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,141)(99,142)(100,143)(101,138)(102,139)(103,140)(104,131)(105,132)(106,133)(107,134)(108,135)(109,136)(110,137)(121,158)(122,159)(123,160)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157), (1,47,27,40)(2,48,28,31)(3,49,29,32)(4,50,30,33)(5,41,21,34)(6,42,22,35)(7,43,23,36)(8,44,24,37)(9,45,25,38)(10,46,26,39)(11,140,160,148)(12,131,151,149)(13,132,152,150)(14,133,153,141)(15,134,154,142)(16,135,155,143)(17,136,156,144)(18,137,157,145)(19,138,158,146)(20,139,159,147)(51,81,68,71)(52,82,69,72)(53,83,70,73)(54,84,61,74)(55,85,62,75)(56,86,63,76)(57,87,64,77)(58,88,65,78)(59,89,66,79)(60,90,67,80)(91,111,109,129)(92,112,110,130)(93,113,101,121)(94,114,102,122)(95,115,103,123)(96,116,104,124)(97,117,105,125)(98,118,106,126)(99,119,107,127)(100,120,108,128), (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), (1,95,6,100)(2,94,7,99)(3,93,8,98)(4,92,9,97)(5,91,10,96)(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( (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,61)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,111)(18,112)(19,113)(20,114)(21,59)(22,60)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,86)(32,87)(33,88)(34,89)(35,90)(36,81)(37,82)(38,83)(39,84)(40,85)(41,79)(42,80)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,141)(99,142)(100,143)(101,138)(102,139)(103,140)(104,131)(105,132)(106,133)(107,134)(108,135)(109,136)(110,137)(121,158)(122,159)(123,160)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157), (1,47,27,40)(2,48,28,31)(3,49,29,32)(4,50,30,33)(5,41,21,34)(6,42,22,35)(7,43,23,36)(8,44,24,37)(9,45,25,38)(10,46,26,39)(11,140,160,148)(12,131,151,149)(13,132,152,150)(14,133,153,141)(15,134,154,142)(16,135,155,143)(17,136,156,144)(18,137,157,145)(19,138,158,146)(20,139,159,147)(51,81,68,71)(52,82,69,72)(53,83,70,73)(54,84,61,74)(55,85,62,75)(56,86,63,76)(57,87,64,77)(58,88,65,78)(59,89,66,79)(60,90,67,80)(91,111,109,129)(92,112,110,130)(93,113,101,121)(94,114,102,122)(95,115,103,123)(96,116,104,124)(97,117,105,125)(98,118,106,126)(99,119,107,127)(100,120,108,128), (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), (1,95,6,100)(2,94,7,99)(3,93,8,98)(4,92,9,97)(5,91,10,96)(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([[(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,61),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,111),(18,112),(19,113),(20,114),(21,59),(22,60),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,86),(32,87),(33,88),(34,89),(35,90),(36,81),(37,82),(38,83),(39,84),(40,85),(41,79),(42,80),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(91,144),(92,145),(93,146),(94,147),(95,148),(96,149),(97,150),(98,141),(99,142),(100,143),(101,138),(102,139),(103,140),(104,131),(105,132),(106,133),(107,134),(108,135),(109,136),(110,137),(121,158),(122,159),(123,160),(124,151),(125,152),(126,153),(127,154),(128,155),(129,156),(130,157)], [(1,47,27,40),(2,48,28,31),(3,49,29,32),(4,50,30,33),(5,41,21,34),(6,42,22,35),(7,43,23,36),(8,44,24,37),(9,45,25,38),(10,46,26,39),(11,140,160,148),(12,131,151,149),(13,132,152,150),(14,133,153,141),(15,134,154,142),(16,135,155,143),(17,136,156,144),(18,137,157,145),(19,138,158,146),(20,139,159,147),(51,81,68,71),(52,82,69,72),(53,83,70,73),(54,84,61,74),(55,85,62,75),(56,86,63,76),(57,87,64,77),(58,88,65,78),(59,89,66,79),(60,90,67,80),(91,111,109,129),(92,112,110,130),(93,113,101,121),(94,114,102,122),(95,115,103,123),(96,116,104,124),(97,117,105,125),(98,118,106,126),(99,119,107,127),(100,120,108,128)], [(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)], [(1,95,6,100),(2,94,7,99),(3,93,8,98),(4,92,9,97),(5,91,10,96),(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)]])
C2×C4⋊Dic5 is a maximal subgroup of
C20.31C42 C20.39C42 M4(2)⋊Dic5 (C2×C20)⋊1C8 C10.49(C4×D4) C2.(C4×D20) C4⋊Dic5⋊15C4 C10.52(C4×D4) (C2×Dic5)⋊Q8 C2.(C20⋊Q8) (C2×C20).28D4 (C2×C4).Dic10 C10.(C4⋊Q8) D10⋊3(C4⋊C4) C10.55(C4×D4) (C2×C4).21D20 (C2×C20).33D4 C23.34D20 C23.35D20 C23.38D20 C22.D40 C20⋊7(C4⋊C4) (C2×C20)⋊10Q8 C42⋊8Dic5 C42⋊9Dic5 (C2×C4)⋊6D20 C24.6D10 C24.7D10 C24.47D10 C24.8D10 C23.14D20 C24.16D10 C20⋊4(C4⋊C4) C4⋊C4×Dic5 C20⋊5(C4⋊C4) C20.48(C4⋊C4) C4⋊C4⋊5Dic5 (C2×C20).53D4 (C2×C20).54D4 C20⋊6(C4⋊C4) (C2×C20).55D4 D10⋊4(C4⋊C4) (C2×C20).56D4 C20.64(C4⋊C4) (C2×C10).D8 C4⋊D4.D5 C22⋊Q8.D5 (C2×C10).Q16 C23.47D20 C23.49D20 C24.64D10 C24.19D10 (Q8×C10)⋊17C4 C4○D4⋊Dic5 C2×C4×Dic10 C2×C4×D20 C2×D5×C4⋊C4 C42.90D10 C42.91D10 C42.105D10 D4⋊6Dic10 D4⋊6D20 C42.119D10 C10.732- 1+4 C10.1152+ 1+4 C10.1182+ 1+4 C10.772- 1+4 C10.852- 1+4 C2×D4×Dic5 C2×Q8×Dic5 C10.1062- 1+4 C10.1472+ 1+4
C2×C4⋊Dic5 is a maximal quotient of
C20⋊13M4(2) C42⋊8Dic5 C42⋊9Dic5 C24.47D10 C20⋊6(C4⋊C4) C42.43D10 C23.22D20 C23.47D20 M4(2).Dic5 C24.64D10
52 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | D5 | Dic5 | D10 | D10 | Dic10 | D20 |
| kernel | C2×C4⋊Dic5 | C4⋊Dic5 | C22×Dic5 | C22×C20 | C2×C20 | C2×C10 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 2 | 2 | 2 | 8 | 4 | 2 | 8 | 8 |
Matrix representation of C2×C4⋊Dic5 ►in GL6(𝔽41)
| 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 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 | 32 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 32 | 18 |
| 14 | 2 | 0 | 0 | 0 | 0 |
| 4 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 39 | 0 | 0 |
| 0 | 0 | 37 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 40 |
| 0 | 0 | 0 | 0 | 9 | 16 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,1,0,0,0,0,0,9],[35,40,0,0,0,0,1,0,0,0,0,0,0,0,35,40,0,0,0,0,1,0,0,0,0,0,0,0,16,32,0,0,0,0,0,18],[14,4,0,0,0,0,2,27,0,0,0,0,0,0,27,37,0,0,0,0,39,14,0,0,0,0,0,0,25,9,0,0,0,0,40,16] >;
C2×C4⋊Dic5 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes {\rm Dic}_5 % in TeX
G:=Group("C2xC4:Dic5"); // GroupNames label
G:=SmallGroup(160,146);
// by ID
G=gap.SmallGroup(160,146);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,362,86,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^10=1,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations