metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.2C8, Dic10.2C8, M4(2).4F5, C5⋊D4.C8, D10.(C2×C8), D5⋊C16⋊5C2, C5⋊1(D4○C16), C20.8(C2×C8), C8.18(C2×F5), C40.18(C2×C4), C4.5(D5⋊C8), C4○D20.2C4, C8⋊D5.2C4, C5⋊C16.2C22, C8.F5⋊4C2, C20.C8⋊3C2, Dic5.2(C2×C8), C4.51(C22×F5), C10.11(C22×C8), C20.91(C22×C4), C5⋊2C8.38C23, C22.1(D5⋊C8), (C8×D5).35C22, (C5×M4(2)).4C4, D20.2C4.3C2, (C2×C5⋊C16)⋊3C2, (C2×C10).1(C2×C8), C2.12(C2×D5⋊C8), (C2×C4).75(C2×F5), (C2×C20).44(C2×C4), C5⋊2C8.20(C2×C4), (C4×D5).45(C2×C4), (C2×C5⋊2C8).188C22, SmallGroup(320,1063)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 226 in 84 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, D5 [×2], C10, C10, C16 [×4], C2×C8 [×3], M4(2), M4(2) [×2], C4○D4, Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C2×C16 [×3], M5(2) [×3], C8○D4, C5⋊2C8 [×2], C40 [×2], Dic10, C4×D5 [×2], D20, C5⋊D4 [×2], C2×C20, D4○C16, C5⋊C16 [×2], C5⋊C16 [×2], C8×D5 [×2], C8⋊D5 [×2], C2×C5⋊2C8, C5×M4(2), C4○D20, D5⋊C16 [×2], C8.F5 [×2], C2×C5⋊C16, C20.C8, D20.2C4, Dic10.C8
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, F5, C22×C8, C2×F5 [×3], D4○C16, D5⋊C8 [×2], C22×F5, C2×D5⋊C8, Dic10.C8
Generators and relations
G = < a,b,c | a20=1, b2=c8=a10, bab-1=a-1, cac-1=a13, bc=cb >
►On 160 points
Generators in S160
(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)
(1 139 11 129)(2 138 12 128)(3 137 13 127)(4 136 14 126)(5 135 15 125)(6 134 16 124)(7 133 17 123)(8 132 18 122)(9 131 19 121)(10 130 20 140)(21 98 31 88)(22 97 32 87)(23 96 33 86)(24 95 34 85)(25 94 35 84)(26 93 36 83)(27 92 37 82)(28 91 38 81)(29 90 39 100)(30 89 40 99)(41 160 51 150)(42 159 52 149)(43 158 53 148)(44 157 54 147)(45 156 55 146)(46 155 56 145)(47 154 57 144)(48 153 58 143)(49 152 59 142)(50 151 60 141)(61 101 71 111)(62 120 72 110)(63 119 73 109)(64 118 74 108)(65 117 75 107)(66 116 76 106)(67 115 77 105)(68 114 78 104)(69 113 79 103)(70 112 80 102)
(1 63 143 89 134 114 43 35 11 73 153 99 124 104 53 25)(2 80 152 82 135 111 52 28 12 70 142 92 125 101 42 38)(3 77 141 95 136 108 41 21 13 67 151 85 126 118 51 31)(4 74 150 88 137 105 50 34 14 64 160 98 127 115 60 24)(5 71 159 81 138 102 59 27 15 61 149 91 128 112 49 37)(6 68 148 94 139 119 48 40 16 78 158 84 129 109 58 30)(7 65 157 87 140 116 57 33 17 75 147 97 130 106 47 23)(8 62 146 100 121 113 46 26 18 72 156 90 131 103 56 36)(9 79 155 93 122 110 55 39 19 69 145 83 132 120 45 29)(10 76 144 86 123 107 44 32 20 66 154 96 133 117 54 22)
G:=sub<Sym(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), (1,139,11,129)(2,138,12,128)(3,137,13,127)(4,136,14,126)(5,135,15,125)(6,134,16,124)(7,133,17,123)(8,132,18,122)(9,131,19,121)(10,130,20,140)(21,98,31,88)(22,97,32,87)(23,96,33,86)(24,95,34,85)(25,94,35,84)(26,93,36,83)(27,92,37,82)(28,91,38,81)(29,90,39,100)(30,89,40,99)(41,160,51,150)(42,159,52,149)(43,158,53,148)(44,157,54,147)(45,156,55,146)(46,155,56,145)(47,154,57,144)(48,153,58,143)(49,152,59,142)(50,151,60,141)(61,101,71,111)(62,120,72,110)(63,119,73,109)(64,118,74,108)(65,117,75,107)(66,116,76,106)(67,115,77,105)(68,114,78,104)(69,113,79,103)(70,112,80,102), (1,63,143,89,134,114,43,35,11,73,153,99,124,104,53,25)(2,80,152,82,135,111,52,28,12,70,142,92,125,101,42,38)(3,77,141,95,136,108,41,21,13,67,151,85,126,118,51,31)(4,74,150,88,137,105,50,34,14,64,160,98,127,115,60,24)(5,71,159,81,138,102,59,27,15,61,149,91,128,112,49,37)(6,68,148,94,139,119,48,40,16,78,158,84,129,109,58,30)(7,65,157,87,140,116,57,33,17,75,147,97,130,106,47,23)(8,62,146,100,121,113,46,26,18,72,156,90,131,103,56,36)(9,79,155,93,122,110,55,39,19,69,145,83,132,120,45,29)(10,76,144,86,123,107,44,32,20,66,154,96,133,117,54,22)>;
G:=Group( (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), (1,139,11,129)(2,138,12,128)(3,137,13,127)(4,136,14,126)(5,135,15,125)(6,134,16,124)(7,133,17,123)(8,132,18,122)(9,131,19,121)(10,130,20,140)(21,98,31,88)(22,97,32,87)(23,96,33,86)(24,95,34,85)(25,94,35,84)(26,93,36,83)(27,92,37,82)(28,91,38,81)(29,90,39,100)(30,89,40,99)(41,160,51,150)(42,159,52,149)(43,158,53,148)(44,157,54,147)(45,156,55,146)(46,155,56,145)(47,154,57,144)(48,153,58,143)(49,152,59,142)(50,151,60,141)(61,101,71,111)(62,120,72,110)(63,119,73,109)(64,118,74,108)(65,117,75,107)(66,116,76,106)(67,115,77,105)(68,114,78,104)(69,113,79,103)(70,112,80,102), (1,63,143,89,134,114,43,35,11,73,153,99,124,104,53,25)(2,80,152,82,135,111,52,28,12,70,142,92,125,101,42,38)(3,77,141,95,136,108,41,21,13,67,151,85,126,118,51,31)(4,74,150,88,137,105,50,34,14,64,160,98,127,115,60,24)(5,71,159,81,138,102,59,27,15,61,149,91,128,112,49,37)(6,68,148,94,139,119,48,40,16,78,158,84,129,109,58,30)(7,65,157,87,140,116,57,33,17,75,147,97,130,106,47,23)(8,62,146,100,121,113,46,26,18,72,156,90,131,103,56,36)(9,79,155,93,122,110,55,39,19,69,145,83,132,120,45,29)(10,76,144,86,123,107,44,32,20,66,154,96,133,117,54,22) );
G=PermutationGroup([(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)], [(1,139,11,129),(2,138,12,128),(3,137,13,127),(4,136,14,126),(5,135,15,125),(6,134,16,124),(7,133,17,123),(8,132,18,122),(9,131,19,121),(10,130,20,140),(21,98,31,88),(22,97,32,87),(23,96,33,86),(24,95,34,85),(25,94,35,84),(26,93,36,83),(27,92,37,82),(28,91,38,81),(29,90,39,100),(30,89,40,99),(41,160,51,150),(42,159,52,149),(43,158,53,148),(44,157,54,147),(45,156,55,146),(46,155,56,145),(47,154,57,144),(48,153,58,143),(49,152,59,142),(50,151,60,141),(61,101,71,111),(62,120,72,110),(63,119,73,109),(64,118,74,108),(65,117,75,107),(66,116,76,106),(67,115,77,105),(68,114,78,104),(69,113,79,103),(70,112,80,102)], [(1,63,143,89,134,114,43,35,11,73,153,99,124,104,53,25),(2,80,152,82,135,111,52,28,12,70,142,92,125,101,42,38),(3,77,141,95,136,108,41,21,13,67,151,85,126,118,51,31),(4,74,150,88,137,105,50,34,14,64,160,98,127,115,60,24),(5,71,159,81,138,102,59,27,15,61,149,91,128,112,49,37),(6,68,148,94,139,119,48,40,16,78,158,84,129,109,58,30),(7,65,157,87,140,116,57,33,17,75,147,97,130,106,47,23),(8,62,146,100,121,113,46,26,18,72,156,90,131,103,56,36),(9,79,155,93,122,110,55,39,19,69,145,83,132,120,45,29),(10,76,144,86,123,107,44,32,20,66,154,96,133,117,54,22)])
Matrix representation ►G ⊆ GL6(𝔽241)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 1 |
| 0 | 0 | 0 | 240 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 115 | 0 | 0 | 0 | 0 | 0 |
| 0 | 115 | 0 | 0 | 0 | 0 |
| 0 | 0 | 227 | 14 | 145 | 0 |
| 0 | 0 | 131 | 14 | 0 | 227 |
| 0 | 0 | 227 | 0 | 14 | 131 |
| 0 | 0 | 0 | 145 | 14 | 227 |
G:=sub<GL(6,GF(241))| [0,15,0,0,0,0,16,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,1,1,1],[0,4,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0],[115,0,0,0,0,0,0,115,0,0,0,0,0,0,227,131,227,0,0,0,14,14,0,145,0,0,145,0,14,14,0,0,0,227,131,227] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 16A | ··· | 16H | 16I | ··· | 16T | 20A | 20B | 20C | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 16 | ··· | 16 | 16 | ··· | 16 | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 10 | 10 | 1 | 1 | 2 | 10 | 10 | 4 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 8 | 5 | ··· | 5 | 10 | ··· | 10 | 4 | 4 | 8 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | D4○C16 | F5 | C2×F5 | C2×F5 | D5⋊C8 | D5⋊C8 | Dic10.C8 |
| kernel | Dic10.C8 | D5⋊C16 | C8.F5 | C2×C5⋊C16 | C20.C8 | D20.2C4 | C8⋊D5 | C5×M4(2) | C4○D20 | Dic10 | D20 | C5⋊D4 | C5 | M4(2) | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 8 | 8 | 1 | 2 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
Dic_{10}.C_8 % in TeX
G:=Group("Dic10.C8"); // GroupNames label
G:=SmallGroup(320,1063);
// by ID
G=gap.SmallGroup(320,1063);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,80,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=1,b^2=c^8=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^13,b*c=c*b>;
// generators/relations
×
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ℤ
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