direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×Dic5, C40⋊7C4, C10.8C42, C5⋊4(C4×C8), C5⋊2C8⋊9C4, C2.2(C8×D5), (C2×C8).10D5, C4.20(C4×D5), C10.12(C2×C8), (C2×C40).11C2, C20.61(C2×C4), (C2×C4).90D10, C2.2(C4×Dic5), C22.8(C4×D5), C4.12(C2×Dic5), (C4×Dic5).14C2, (C2×Dic5).15C4, (C2×C20).104C22, (C2×C5⋊2C8).14C2, (C2×C10).29(C2×C4), SmallGroup(160,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C8×Dic5 |
Generators and relations for C8×Dic5
G = < a,b,c | a8=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C8×Dic5
►Regular action on 160 points
Generators in S160
(1 90 48 74 39 63 14 55)(2 81 49 75 40 64 15 56)(3 82 50 76 31 65 16 57)(4 83 41 77 32 66 17 58)(5 84 42 78 33 67 18 59)(6 85 43 79 34 68 19 60)(7 86 44 80 35 69 20 51)(8 87 45 71 36 70 11 52)(9 88 46 72 37 61 12 53)(10 89 47 73 38 62 13 54)(21 127 156 111 131 96 144 109)(22 128 157 112 132 97 145 110)(23 129 158 113 133 98 146 101)(24 130 159 114 134 99 147 102)(25 121 160 115 135 100 148 103)(26 122 151 116 136 91 149 104)(27 123 152 117 137 92 150 105)(28 124 153 118 138 93 141 106)(29 125 154 119 139 94 142 107)(30 126 155 120 140 95 143 108)
(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)
(1 104 6 109)(2 103 7 108)(3 102 8 107)(4 101 9 106)(5 110 10 105)(11 94 16 99)(12 93 17 98)(13 92 18 97)(14 91 19 96)(15 100 20 95)(21 90 26 85)(22 89 27 84)(23 88 28 83)(24 87 29 82)(25 86 30 81)(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)(51 143 56 148)(52 142 57 147)(53 141 58 146)(54 150 59 145)(55 149 60 144)(61 138 66 133)(62 137 67 132)(63 136 68 131)(64 135 69 140)(65 134 70 139)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)
G:=sub<Sym(160)| (1,90,48,74,39,63,14,55)(2,81,49,75,40,64,15,56)(3,82,50,76,31,65,16,57)(4,83,41,77,32,66,17,58)(5,84,42,78,33,67,18,59)(6,85,43,79,34,68,19,60)(7,86,44,80,35,69,20,51)(8,87,45,71,36,70,11,52)(9,88,46,72,37,61,12,53)(10,89,47,73,38,62,13,54)(21,127,156,111,131,96,144,109)(22,128,157,112,132,97,145,110)(23,129,158,113,133,98,146,101)(24,130,159,114,134,99,147,102)(25,121,160,115,135,100,148,103)(26,122,151,116,136,91,149,104)(27,123,152,117,137,92,150,105)(28,124,153,118,138,93,141,106)(29,125,154,119,139,94,142,107)(30,126,155,120,140,95,143,108), (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), (1,104,6,109)(2,103,7,108)(3,102,8,107)(4,101,9,106)(5,110,10,105)(11,94,16,99)(12,93,17,98)(13,92,18,97)(14,91,19,96)(15,100,20,95)(21,90,26,85)(22,89,27,84)(23,88,28,83)(24,87,29,82)(25,86,30,81)(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)(51,143,56,148)(52,142,57,147)(53,141,58,146)(54,150,59,145)(55,149,60,144)(61,138,66,133)(62,137,67,132)(63,136,68,131)(64,135,69,140)(65,134,70,139)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155)>;
G:=Group( (1,90,48,74,39,63,14,55)(2,81,49,75,40,64,15,56)(3,82,50,76,31,65,16,57)(4,83,41,77,32,66,17,58)(5,84,42,78,33,67,18,59)(6,85,43,79,34,68,19,60)(7,86,44,80,35,69,20,51)(8,87,45,71,36,70,11,52)(9,88,46,72,37,61,12,53)(10,89,47,73,38,62,13,54)(21,127,156,111,131,96,144,109)(22,128,157,112,132,97,145,110)(23,129,158,113,133,98,146,101)(24,130,159,114,134,99,147,102)(25,121,160,115,135,100,148,103)(26,122,151,116,136,91,149,104)(27,123,152,117,137,92,150,105)(28,124,153,118,138,93,141,106)(29,125,154,119,139,94,142,107)(30,126,155,120,140,95,143,108), (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), (1,104,6,109)(2,103,7,108)(3,102,8,107)(4,101,9,106)(5,110,10,105)(11,94,16,99)(12,93,17,98)(13,92,18,97)(14,91,19,96)(15,100,20,95)(21,90,26,85)(22,89,27,84)(23,88,28,83)(24,87,29,82)(25,86,30,81)(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)(51,143,56,148)(52,142,57,147)(53,141,58,146)(54,150,59,145)(55,149,60,144)(61,138,66,133)(62,137,67,132)(63,136,68,131)(64,135,69,140)(65,134,70,139)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155) );
G=PermutationGroup([[(1,90,48,74,39,63,14,55),(2,81,49,75,40,64,15,56),(3,82,50,76,31,65,16,57),(4,83,41,77,32,66,17,58),(5,84,42,78,33,67,18,59),(6,85,43,79,34,68,19,60),(7,86,44,80,35,69,20,51),(8,87,45,71,36,70,11,52),(9,88,46,72,37,61,12,53),(10,89,47,73,38,62,13,54),(21,127,156,111,131,96,144,109),(22,128,157,112,132,97,145,110),(23,129,158,113,133,98,146,101),(24,130,159,114,134,99,147,102),(25,121,160,115,135,100,148,103),(26,122,151,116,136,91,149,104),(27,123,152,117,137,92,150,105),(28,124,153,118,138,93,141,106),(29,125,154,119,139,94,142,107),(30,126,155,120,140,95,143,108)], [(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)], [(1,104,6,109),(2,103,7,108),(3,102,8,107),(4,101,9,106),(5,110,10,105),(11,94,16,99),(12,93,17,98),(13,92,18,97),(14,91,19,96),(15,100,20,95),(21,90,26,85),(22,89,27,84),(23,88,28,83),(24,87,29,82),(25,86,30,81),(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),(51,143,56,148),(52,142,57,147),(53,141,58,146),(54,150,59,145),(55,149,60,144),(61,138,66,133),(62,137,67,132),(63,136,68,131),(64,135,69,140),(65,134,70,139),(71,154,76,159),(72,153,77,158),(73,152,78,157),(74,151,79,156),(75,160,80,155)]])
C8×Dic5 is a maximal subgroup of
C40.88D4 C80⋊17C4 C40.9Q8 C40⋊C8 C20.31M4(2) C40⋊2C8 C40⋊1C8 C20.26M4(2) Dic5.13D8 Dic5⋊C16 C40.C8 C10.M5(2) C40.1C8 D5×C4×C8 D10.5C42 C40⋊Q8 D10.6C42 D10.7C42 Dic5.14M4(2) C40⋊8C4⋊C2 C5⋊5(C8×D4) Dic5⋊2M4(2) Dic5⋊4D8 Dic5⋊6SD16 Dic5.5D8 (C8×Dic5)⋊C2 Dic5⋊7SD16 Dic5⋊4Q16 Dic5.3Q16 Q8⋊Dic5⋊C2 Dic5.5M4(2) Dic10⋊5C8 C42.200D10 C42.31D10 Dic5⋊8SD16 C40⋊5Q8 C8.8Dic10 D40⋊12C4 Dic5⋊5Q16 C40⋊2Q8 C8.6Dic10 D40⋊13C4 C20.42C42 C20.37C42 C40⋊18D4 C40.93D4 C40⋊5D4 C40.22D4 C40.43D4 C40⋊15D4 C40.26D4 C40.28D4 D8⋊5Dic5 Dic15⋊4C8
C8×Dic5 is a maximal quotient of
C42.279D10 C80⋊17C4 (C2×C40)⋊15C4 Dic15⋊4C8
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D5 | Dic5 | D10 | C4×D5 | C4×D5 | C8×D5 |
| kernel | C8×Dic5 | C2×C5⋊2C8 | C4×Dic5 | C2×C40 | C5⋊2C8 | C40 | C2×Dic5 | Dic5 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 16 | 2 | 4 | 2 | 4 | 4 | 16 |
Matrix representation of C8×Dic5 ►in GL3(𝔽41) generated by
| 1 | 0 | 0 |
| 0 | 14 | 0 |
| 0 | 0 | 14 |
| 40 | 0 | 0 |
| 0 | 0 | 40 |
| 0 | 1 | 35 |
| 32 | 0 | 0 |
| 0 | 2 | 15 |
| 0 | 27 | 39 |
G:=sub<GL(3,GF(41))| [1,0,0,0,14,0,0,0,14],[40,0,0,0,0,1,0,40,35],[32,0,0,0,2,27,0,15,39] >;
C8×Dic5 in GAP, Magma, Sage, TeX
C_8\times {\rm Dic}_5 % in TeX
G:=Group("C8xDic5"); // GroupNames label
G:=SmallGroup(160,20);
// by ID
G=gap.SmallGroup(160,20);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,55,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^8=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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