metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×Dic5)⋊1C8, C22⋊C8.2D5, C22.4(C8×D5), (C2×C4).108D20, (C2×C20).439D4, (C22×C4).2D10, C23.41(C4×D5), C2.6(D10⋊1C8), C10.19(C22⋊C8), C10.26(C23⋊C4), C22.4(C8⋊D5), (C2×C10).11M4(2), (C22×Dic5).2C4, C20.55D4.11C2, C10.7(C4.10D4), C2.1(C4.12D20), (C22×C20).323C22, C2.2(C23.1D10), C5⋊3(C22.M4(2)), C22.33(D10⋊C4), (C2×C10).17(C2×C8), (C5×C22⋊C8).2C2, (C2×C4).210(C5⋊D4), (C22×C10).95(C2×C4), (C2×C10.D4).26C2, (C2×C10).105(C22⋊C4), SmallGroup(320,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×Dic5)⋊C8
G = < a,b,c,d | a2=b10=d8=1, c2=b5, ab=ba, ac=ca, dad-1=ab5, cbc-1=b-1, bd=db, dcd-1=ab5c >
Subgroups: 278 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C2×C8, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C22⋊C8, C22⋊C8, C2×C4⋊C4, C5⋊2C8, C40, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22.M4(2), C2×C5⋊2C8, C10.D4, C2×C40, C22×Dic5, C22×C20, C20.55D4, C5×C22⋊C8, C2×C10.D4, (C2×Dic5)⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D5, C22⋊C4, C2×C8, M4(2), D10, C22⋊C8, C23⋊C4, C4.10D4, C4×D5, D20, C5⋊D4, C22.M4(2), C8×D5, C8⋊D5, D10⋊C4, C23.1D10, D10⋊1C8, C4.12D20, (C2×Dic5)⋊C8
►On 160 points
Generators in S160
(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)(121 126)(122 127)(123 128)(124 129)(125 130)(131 136)(132 137)(133 138)(134 139)(135 140)(141 146)(142 147)(143 148)(144 149)(145 150)(151 156)(152 157)(153 158)(154 159)(155 160)
(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 84 6 89)(2 83 7 88)(3 82 8 87)(4 81 9 86)(5 90 10 85)(11 94 16 99)(12 93 17 98)(13 92 18 97)(14 91 19 96)(15 100 20 95)(21 104 26 109)(22 103 27 108)(23 102 28 107)(24 101 29 106)(25 110 30 105)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 124 46 129)(42 123 47 128)(43 122 48 127)(44 121 49 126)(45 130 50 125)(51 134 56 139)(52 133 57 138)(53 132 58 137)(54 131 59 136)(55 140 60 135)(61 144 66 149)(62 143 67 148)(63 142 68 147)(64 141 69 146)(65 150 70 145)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)
(1 71 31 51 11 61 21 41)(2 72 32 52 12 62 22 42)(3 73 33 53 13 63 23 43)(4 74 34 54 14 64 24 44)(5 75 35 55 15 65 25 45)(6 76 36 56 16 66 26 46)(7 77 37 57 17 67 27 47)(8 78 38 58 18 68 28 48)(9 79 39 59 19 69 29 49)(10 80 40 60 20 70 30 50)(81 156 116 131 91 146 106 121)(82 157 117 132 92 147 107 122)(83 158 118 133 93 148 108 123)(84 159 119 134 94 149 109 124)(85 160 120 135 95 150 110 125)(86 151 111 136 96 141 101 126)(87 152 112 137 97 142 102 127)(88 153 113 138 98 143 103 128)(89 154 114 139 99 144 104 129)(90 155 115 140 100 145 105 130)
G:=sub<Sym(160)| (41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80)(121,126)(122,127)(123,128)(124,129)(125,130)(131,136)(132,137)(133,138)(134,139)(135,140)(141,146)(142,147)(143,148)(144,149)(145,150)(151,156)(152,157)(153,158)(154,159)(155,160), (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), (1,84,6,89)(2,83,7,88)(3,82,8,87)(4,81,9,86)(5,90,10,85)(11,94,16,99)(12,93,17,98)(13,92,18,97)(14,91,19,96)(15,100,20,95)(21,104,26,109)(22,103,27,108)(23,102,28,107)(24,101,29,106)(25,110,30,105)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,124,46,129)(42,123,47,128)(43,122,48,127)(44,121,49,126)(45,130,50,125)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,144,66,149)(62,143,67,148)(63,142,68,147)(64,141,69,146)(65,150,70,145)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155), (1,71,31,51,11,61,21,41)(2,72,32,52,12,62,22,42)(3,73,33,53,13,63,23,43)(4,74,34,54,14,64,24,44)(5,75,35,55,15,65,25,45)(6,76,36,56,16,66,26,46)(7,77,37,57,17,67,27,47)(8,78,38,58,18,68,28,48)(9,79,39,59,19,69,29,49)(10,80,40,60,20,70,30,50)(81,156,116,131,91,146,106,121)(82,157,117,132,92,147,107,122)(83,158,118,133,93,148,108,123)(84,159,119,134,94,149,109,124)(85,160,120,135,95,150,110,125)(86,151,111,136,96,141,101,126)(87,152,112,137,97,142,102,127)(88,153,113,138,98,143,103,128)(89,154,114,139,99,144,104,129)(90,155,115,140,100,145,105,130)>;
G:=Group( (41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80)(121,126)(122,127)(123,128)(124,129)(125,130)(131,136)(132,137)(133,138)(134,139)(135,140)(141,146)(142,147)(143,148)(144,149)(145,150)(151,156)(152,157)(153,158)(154,159)(155,160), (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), (1,84,6,89)(2,83,7,88)(3,82,8,87)(4,81,9,86)(5,90,10,85)(11,94,16,99)(12,93,17,98)(13,92,18,97)(14,91,19,96)(15,100,20,95)(21,104,26,109)(22,103,27,108)(23,102,28,107)(24,101,29,106)(25,110,30,105)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,124,46,129)(42,123,47,128)(43,122,48,127)(44,121,49,126)(45,130,50,125)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,144,66,149)(62,143,67,148)(63,142,68,147)(64,141,69,146)(65,150,70,145)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155), (1,71,31,51,11,61,21,41)(2,72,32,52,12,62,22,42)(3,73,33,53,13,63,23,43)(4,74,34,54,14,64,24,44)(5,75,35,55,15,65,25,45)(6,76,36,56,16,66,26,46)(7,77,37,57,17,67,27,47)(8,78,38,58,18,68,28,48)(9,79,39,59,19,69,29,49)(10,80,40,60,20,70,30,50)(81,156,116,131,91,146,106,121)(82,157,117,132,92,147,107,122)(83,158,118,133,93,148,108,123)(84,159,119,134,94,149,109,124)(85,160,120,135,95,150,110,125)(86,151,111,136,96,141,101,126)(87,152,112,137,97,142,102,127)(88,153,113,138,98,143,103,128)(89,154,114,139,99,144,104,129)(90,155,115,140,100,145,105,130) );
G=PermutationGroup([[(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80),(121,126),(122,127),(123,128),(124,129),(125,130),(131,136),(132,137),(133,138),(134,139),(135,140),(141,146),(142,147),(143,148),(144,149),(145,150),(151,156),(152,157),(153,158),(154,159),(155,160)], [(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)], [(1,84,6,89),(2,83,7,88),(3,82,8,87),(4,81,9,86),(5,90,10,85),(11,94,16,99),(12,93,17,98),(13,92,18,97),(14,91,19,96),(15,100,20,95),(21,104,26,109),(22,103,27,108),(23,102,28,107),(24,101,29,106),(25,110,30,105),(31,114,36,119),(32,113,37,118),(33,112,38,117),(34,111,39,116),(35,120,40,115),(41,124,46,129),(42,123,47,128),(43,122,48,127),(44,121,49,126),(45,130,50,125),(51,134,56,139),(52,133,57,138),(53,132,58,137),(54,131,59,136),(55,140,60,135),(61,144,66,149),(62,143,67,148),(63,142,68,147),(64,141,69,146),(65,150,70,145),(71,154,76,159),(72,153,77,158),(73,152,78,157),(74,151,79,156),(75,160,80,155)], [(1,71,31,51,11,61,21,41),(2,72,32,52,12,62,22,42),(3,73,33,53,13,63,23,43),(4,74,34,54,14,64,24,44),(5,75,35,55,15,65,25,45),(6,76,36,56,16,66,26,46),(7,77,37,57,17,67,27,47),(8,78,38,58,18,68,28,48),(9,79,39,59,19,69,29,49),(10,80,40,60,20,70,30,50),(81,156,116,131,91,146,106,121),(82,157,117,132,92,147,107,122),(83,158,118,133,93,148,108,123),(84,159,119,134,94,149,109,124),(85,160,120,135,95,150,110,125),(86,151,111,136,96,141,101,126),(87,152,112,137,97,142,102,127),(88,153,113,138,98,143,103,128),(89,154,114,139,99,144,104,129),(90,155,115,140,100,145,105,130)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C8 | D4 | D5 | M4(2) | D10 | D20 | C5⋊D4 | C4×D5 | C8×D5 | C8⋊D5 | C23⋊C4 | C4.10D4 | C23.1D10 | C4.12D20 |
| kernel | (C2×Dic5)⋊C8 | C20.55D4 | C5×C22⋊C8 | C2×C10.D4 | C22×Dic5 | C2×Dic5 | C2×C20 | C22⋊C8 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of (C2×Dic5)⋊C8 ►in GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 13 | 1 | 0 | 40 |
| 3 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 18 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 34 | 11 | 26 |
| 0 | 0 | 0 | 0 | 26 | 2 | 30 | 30 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 17 | 39 |
| 0 | 0 | 0 | 0 | 35 | 1 | 9 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 | 3 | 40 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,16,13,0,0,0,0,0,1,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[3,34,0,0,0,0,0,0,34,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[18,6,0,0,0,0,0,0,35,23,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,27,38,0,26,0,0,0,0,11,14,34,2,0,0,0,0,0,0,11,30,0,0,0,0,0,0,26,30],[0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,32,0,35,6,0,0,0,0,0,1,1,0,0,0,0,0,37,17,9,3,0,0,0,0,0,39,0,40] >;
(C2×Dic5)⋊C8 in GAP, Magma, Sage, TeX
(C_2\times {\rm Dic}_5)\rtimes C_8 % in TeX
G:=Group("(C2xDic5):C8"); // GroupNames label
G:=SmallGroup(320,27);
// by ID
G=gap.SmallGroup(320,27);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,758,100,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=d^8=1,c^2=b^5,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^5,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a*b^5*c>;
// generators/relations