metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.8D10, (C8×Dic5)⋊20C2, D4⋊C4.1D5, (C2×D4).22D10, C20.6(C4○D4), C10.Q16⋊3C2, (C2×C8).200D10, C4.Dic10⋊4C2, C4.23(C4○D20), C10.38(C4○D8), C2.8(D8⋊3D5), D4⋊Dic5.6C2, C22.170(D4×D5), C4.49(D4⋊2D5), C20.44D4⋊17C2, (C2×C40).181C22, (C2×C20).208C23, (C2×Dic5).132D4, C20.17D4.6C2, (D4×C10).29C22, C4⋊Dic5.66C22, C2.9(SD16⋊3D5), C10.25(C4.4D4), C5⋊2(C42.78C22), (C2×Dic10).58C22, (C4×Dic5).254C22, C2.15(Dic5.5D4), (C2×C10).221(C2×D4), (C5×C4⋊C4).13C22, (C5×D4⋊C4).10C2, (C2×C4).315(C22×D5), (C2×C5⋊2C8).218C22, SmallGroup(320,395)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C8×Dic5)⋊C2
G = < a,b,c,d | a8=b10=d2=1, c2=b5, ab=ba, ac=ca, dad=a-1b5, cbc-1=b-1, bd=db, dcd=a4b5c >
Subgroups: 350 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C4×C8, D4⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C42.C2, C5⋊2C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C42.78C22, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C23.D5, C5×C4⋊C4, C2×C40, C2×Dic10, D4×C10, C10.Q16, C8×Dic5, C20.44D4, D4⋊Dic5, C5×D4⋊C4, C4.Dic10, C20.17D4, (C8×Dic5)⋊C2
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C4○D8, C22×D5, C42.78C22, C4○D20, D4×D5, D4⋊2D5, Dic5.5D4, D8⋊3D5, SD16⋊3D5, (C8×Dic5)⋊C2
►On 160 points
Generators in S160
(1 54 39 85 27 67 42 79)(2 55 40 86 28 68 43 80)(3 56 31 87 29 69 44 71)(4 57 32 88 30 70 45 72)(5 58 33 89 21 61 46 73)(6 59 34 90 22 62 47 74)(7 60 35 81 23 63 48 75)(8 51 36 82 24 64 49 76)(9 52 37 83 25 65 50 77)(10 53 38 84 26 66 41 78)(11 103 143 128 156 91 131 116)(12 104 144 129 157 92 132 117)(13 105 145 130 158 93 133 118)(14 106 146 121 159 94 134 119)(15 107 147 122 160 95 135 120)(16 108 148 123 151 96 136 111)(17 109 149 124 152 97 137 112)(18 110 150 125 153 98 138 113)(19 101 141 126 154 99 139 114)(20 102 142 127 155 100 140 115)
(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 91 6 96)(2 100 7 95)(3 99 8 94)(4 98 9 93)(5 97 10 92)(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)
(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 41)(39 42)(40 43)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 81)(69 82)(70 83)(91 108)(92 109)(93 110)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(100 107)(111 116)(112 117)(113 118)(114 119)(115 120)(121 126)(122 127)(123 128)(124 129)(125 130)(141 154)(142 155)(143 156)(144 157)(145 158)(146 159)(147 160)(148 151)(149 152)(150 153)
G:=sub<Sym(160)| (1,54,39,85,27,67,42,79)(2,55,40,86,28,68,43,80)(3,56,31,87,29,69,44,71)(4,57,32,88,30,70,45,72)(5,58,33,89,21,61,46,73)(6,59,34,90,22,62,47,74)(7,60,35,81,23,63,48,75)(8,51,36,82,24,64,49,76)(9,52,37,83,25,65,50,77)(10,53,38,84,26,66,41,78)(11,103,143,128,156,91,131,116)(12,104,144,129,157,92,132,117)(13,105,145,130,158,93,133,118)(14,106,146,121,159,94,134,119)(15,107,147,122,160,95,135,120)(16,108,148,123,151,96,136,111)(17,109,149,124,152,97,137,112)(18,110,150,125,153,98,138,113)(19,101,141,126,154,99,139,114)(20,102,142,127,155,100,140,115), (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), (1,91,6,96)(2,100,7,95)(3,99,8,94)(4,98,9,93)(5,97,10,92)(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), (11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,41)(39,42)(40,43)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,81)(69,82)(70,83)(91,108)(92,109)(93,110)(94,101)(95,102)(96,103)(97,104)(98,105)(99,106)(100,107)(111,116)(112,117)(113,118)(114,119)(115,120)(121,126)(122,127)(123,128)(124,129)(125,130)(141,154)(142,155)(143,156)(144,157)(145,158)(146,159)(147,160)(148,151)(149,152)(150,153)>;
G:=Group( (1,54,39,85,27,67,42,79)(2,55,40,86,28,68,43,80)(3,56,31,87,29,69,44,71)(4,57,32,88,30,70,45,72)(5,58,33,89,21,61,46,73)(6,59,34,90,22,62,47,74)(7,60,35,81,23,63,48,75)(8,51,36,82,24,64,49,76)(9,52,37,83,25,65,50,77)(10,53,38,84,26,66,41,78)(11,103,143,128,156,91,131,116)(12,104,144,129,157,92,132,117)(13,105,145,130,158,93,133,118)(14,106,146,121,159,94,134,119)(15,107,147,122,160,95,135,120)(16,108,148,123,151,96,136,111)(17,109,149,124,152,97,137,112)(18,110,150,125,153,98,138,113)(19,101,141,126,154,99,139,114)(20,102,142,127,155,100,140,115), (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), (1,91,6,96)(2,100,7,95)(3,99,8,94)(4,98,9,93)(5,97,10,92)(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), (11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,41)(39,42)(40,43)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,81)(69,82)(70,83)(91,108)(92,109)(93,110)(94,101)(95,102)(96,103)(97,104)(98,105)(99,106)(100,107)(111,116)(112,117)(113,118)(114,119)(115,120)(121,126)(122,127)(123,128)(124,129)(125,130)(141,154)(142,155)(143,156)(144,157)(145,158)(146,159)(147,160)(148,151)(149,152)(150,153) );
G=PermutationGroup([[(1,54,39,85,27,67,42,79),(2,55,40,86,28,68,43,80),(3,56,31,87,29,69,44,71),(4,57,32,88,30,70,45,72),(5,58,33,89,21,61,46,73),(6,59,34,90,22,62,47,74),(7,60,35,81,23,63,48,75),(8,51,36,82,24,64,49,76),(9,52,37,83,25,65,50,77),(10,53,38,84,26,66,41,78),(11,103,143,128,156,91,131,116),(12,104,144,129,157,92,132,117),(13,105,145,130,158,93,133,118),(14,106,146,121,159,94,134,119),(15,107,147,122,160,95,135,120),(16,108,148,123,151,96,136,111),(17,109,149,124,152,97,137,112),(18,110,150,125,153,98,138,113),(19,101,141,126,154,99,139,114),(20,102,142,127,155,100,140,115)], [(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)], [(1,91,6,96),(2,100,7,95),(3,99,8,94),(4,98,9,93),(5,97,10,92),(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)], [(11,131),(12,132),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,41),(39,42),(40,43),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,84),(62,85),(63,86),(64,87),(65,88),(66,89),(67,90),(68,81),(69,82),(70,83),(91,108),(92,109),(93,110),(94,101),(95,102),(96,103),(97,104),(98,105),(99,106),(100,107),(111,116),(112,117),(113,118),(114,119),(115,120),(121,126),(122,127),(123,128),(124,129),(125,130),(141,154),(142,155),(143,156),(144,157),(145,158),(146,159),(147,160),(148,151),(149,152),(150,153)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 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 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 8 | 10 | 10 | 10 | 10 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4○D20 | D4⋊2D5 | D4×D5 | D8⋊3D5 | SD16⋊3D5 |
| kernel | (C8×Dic5)⋊C2 | C10.Q16 | C8×Dic5 | C20.44D4 | D4⋊Dic5 | C5×D4⋊C4 | C4.Dic10 | C20.17D4 | C2×Dic5 | D4⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C10 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of (C8×Dic5)⋊C2 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 30 |
| 0 | 0 | 0 | 0 | 26 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 8 | 7 | 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 |
| 38 | 33 | 0 | 0 | 0 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 18 | 0 | 0 |
| 0 | 0 | 20 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 22 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,30,26,0,0,0,0,30,0],[40,8,0,0,0,0,40,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[38,1,0,0,0,0,33,3,0,0,0,0,0,0,7,20,0,0,0,0,18,34,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,22,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40] >;
(C8×Dic5)⋊C2 in GAP, Magma, Sage, TeX
(C_8\times {\rm Dic}_5)\rtimes C_2 % in TeX
G:=Group("(C8xDic5):C2"); // GroupNames label
G:=SmallGroup(320,395);
// by ID
G=gap.SmallGroup(320,395);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,1094,135,100,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^10=d^2=1,c^2=b^5,a*b=b*a,a*c=c*a,d*a*d=a^-1*b^5,c*b*c^-1=b^-1,b*d=d*b,d*c*d=a^4*b^5*c>;
// generators/relations